Well, welcome to episode 32. In this episode, we're going to continue talking about sequences and series like we did in the previous episode. However, today we're going to talk about two special sequence and series, those that are called arithmetic sequences and arithmetic series and geometric sequences and series. Let's go to our list of objectives and we'll see what we're going to be doing today. In just a moment, I'll be introducing arithmetic sequences. And then we'll look at two formulas for adding up arithmetic series. And we'll look at several applications of that. Then we'll switch over in the second half of this episode to an introduction to geometric sequences. And there are two formulas that are associated with geometric series. And again, we'll look at some applications of those. OK, now to begin with, let's look at what I mean by an arithmetic sequence. You know, if you're counting like 1, 2, 3, 4, 5, et cetera, you're just listing the numbers and putting commas in between. That would be an example of a sequence. And in particular, that's an example of an arithmetic sequence. Because you notice these numbers are increasing by the same amount. Namely, they're increasing by 1. So when I say 1, 2, 3, 4, my numbers are increasing by 1. So that's the characteristic of an arithmetic sequence. On the other hand, if I were adding up these numbers, like 1 plus 2 plus 3 plus 4, that would be a series, not a sequence, and that would be called an arithmetic sequence. Because again, the numbers are increasing by the same amount. Let's go to the first graphic or the next graphic and we'll look at a formal definition of an arithmetic sequence. A sequence is arithmetic if each pair of consecutive terms differs by the same amount. So I'm going to call the difference between consecutive terms d. So the first term I'll call a. The next term I'll call a plus d. So it differs from the first term by d. And then if I add on another d, I get a plus 2d. It differs from the previous term by d. And if I go to the nth term, I've added on n minus 1d. So right below this, this is a1. This is a2, namely a plus d. This is a3.  So you notice that in the second term, I've added on 1d. In the third term, I've added on 2d's. You notice the number of d's is always one less than the position number, because there wasn't a d in the very first term. So in the nth term, I've added on n minus 1d's. And I'm using the letter d here for difference in that case. So if I write a, I'll be referring to a1. And then, of course, d is the difference. Let's look at some examples of arithmetic sequences in the next graphic. Which of the following are arithmetic sequences? And if it's arithmetic, we're asked to find a and d in the nth term. Well, let's see. In the first example, we have 5, 8, 11, 14, 17. You notice this is a sequence, because I'm just listing the numbers. It's not a series. If it were a series, I'd be adding these numbers together. What appears to be the common difference of the numbers that we've written? Looks like it's 3. So it looks like a and d can be found in this sequence. So it looks like it's arithmetic. And I would say that a is, well, what would you say is a? a would be the first term, so that would be 5. And Stephen said the common difference appears to be 3. And now, what is the nth term? Well, if you remember, the formula that we just saw on the previous graphic said that the nth term is a plus n minus 1 times d. So in this case, that would be 5 plus n minus 1 times 3. Let me just clarify that a little bit so you can see n minus 1 there. Now, if you want, you could leave your answer in this form, or if you wish, you could multiply it out. If I multiply that out, that's 5 plus 3n minus 3. Or when I combine terms, that'll be 3n plus 2. It looks a lot simpler in this form, but I think in this form, we recognize the nth term of the arithmetic sequence. Now, for example, if I substituted in a 1 right there, how much would I get for a sub 1 if I plugged a 1 in right there? Wouldn't that be 0? So how much would this be? 5. It'd be 5. Yeah, I'm sure enough, that is a sub 1. If I substituted in a 2 right here, that'd be 5 plus 1 times 3 would be 8. Yeah, and that's the second term. And et cetera. So this would actually give me any of the terms in the sequence, assuming that this pattern continues. Now, if you look at the formula that I've reduced it to, 3n plus 2, look what happens if I substitute in a 1 in that expression. What will I get? 5. We get 5 again. So again, we get 5 for the first term. And if I substituted in a 2 in this expression, I'll get 8. And I get the second term. So you see, this formula certainly will give me the nth term. And this formula gives me the nth term. So you could write your answer in either of these forms. The latter one is reduced. But I think there's some advantage in leaving the answer in this form, just so that we recognize our general formula. OK, so the first sequence there is an arithmetic sequence. Let's look at the second one in part b. OK, in this case, let's say we start off with 1 tenth, and then we go to 8 hundredths, and then 600ths, and then 400ths. It looks like these numbers are getting smaller. And are they being reduced by the same amount every time? I think so, because you see, if I put a little 0 in here right after that term, then this is 0.10, 0.8, 0.6, or rather, 0.08, 0.06, 0.04. So David, what do you think d is in that sequence? 0.02. 0.02, and it'll be a negative, though, because we're reducing it. Yeah, so what we're doing is we're adding on negative 0.02. And a, the first term, is 0.1. I'll take that 0 back out over here. So it looks like this is an arithmetic sequence if this pattern were to continue. And the nth term, let's see, the term for a sub n would be a plus n minus 1 times d. And that would be 1 tenth plus n minus 1 times negative 0.02. So we're using decimals here. And also, we have a negative value for d, but it's still arithmetic.   Yes, that'll be 0.12 minus 0.02 n. So this is my reduced formula, but somehow this one doesn't look quite as pretty as that last one did in the first example. So you could give your answer this way, or you could give your answer this way for the nth term. OK, so the first two are both arithmetic sequences. Let's go to the third case, or the third example. In this case, we have polynomials. We have a variable included. So we have x, 3x, 5x, 7x. Does it appear that we're adding on the same amount every time? Yeah. Yeah, how much are we adding on? 2x. 2x. So that would make this arithmetic if that's the pattern that continues. And it looks like, in this case, a would be x, d would be 2x, because that's what we're adding on every time. And a sub n, let's see now, a sub n has the formula a plus n minus 1d. You remember the reason this is an n minus 1d is because in the nth term, I've only added on n minus 1d. So it wasn't a d in the first term. So the number of d's that have been added is always 1 less than the position number. And therefore, a sub n would be x plus n minus 1 times 2x. And if I multiply that out, that would be x plus 2nx minus 2x. And that would be, let's see, we need to remove the graphic, I think, up there. There we go. OK. And that would be 2nx minus x. And the last way I'll write this is to factor out the x and put it over here on the right, 2n minus 1 times x. Now in this form, I think we can certainly see the connection between this formula and this sequence. Because you see, if n is an integer, then 2n would be an even integer. So this would be even. 2n minus 1 would be odd. This is an even number minus 1. So this is an odd multiple of x. And that's what I have here, or odd multiples of x. So this certainly seems like a sort of a nice expression to reduce it to. Whereas up here, this seems quite a bit more complicated without the things reduced. But again, this shows us that it's in the standard form for the nth term of an arithmetic sequence. OK. Now let's look at some applications of arithmetic sequences, or at least some problems that pertain to them. And we'll begin with an example that we have on the next graphic. This problem says, suppose the second term of an arithmetic sequence is 14, and the fifth term is 50. What's the seventh term? OK, so the question's about the seventh term. So let's just say that these are the positions of the seven numbers that we have so far up to the seventh term. And the second term of the arithmetic sequence is 14. So I'll put a 14 there. And the fifth term is 50. And we want to know what is the seventh term over here. Well, let's see now. The first term is called A. And after that, I would say that this one is A2, and this one is A5. And the one over here is A sub 7. That's what we're looking for. And I know that A2 has a formula.  Well, here's n minus 1. If n is 2, n minus 1 is 1. So that's just A plus d. I'll just take out the 1 right there. And I know that this is equal to 14. Now what about A sub 5? A sub 5 would be A plus how many d's? 4d. 4d. Yeah, that would be 4d, because I have one less d than I had the position. This is in the fifth position. And that's equal to 50. So you see, what I have here are two equations and two unknowns, two linear equations. We have A, we have d, and we have two linear equations that give their values. So why don't we subtract these two and solve for d? Now, of course, we could use matrix methods. We could use Kramer's rule, but for a 2 by 2, I mean, we might as well just use the methods that we've seen earlier.  So that says that d is equal to what? 12. Yeah, d's equal to 12, exactly. Now, if I plug d in back up here, then that says A plus 12 is 14, and so A is equal to 2. So now I know A, so I know where the series begins, and I know d, these are increases by 12, and we want to find A sub 7. Well, let's just record this information over here on the side, and let's go back and calculate A sub 7. Now, you might say, Dennis, why don't you just fill in all the blanks, starting off here with 2 and add 12 and add 12 and add 12, and eventually you get over here to A to the seventh. What I'm thinking is what if this had been A to the 100th power, or A sub 100, then it would be difficult to write out all those terms. Let's see if we can calculate A sub 7 directly, and I'm thinking A sub 7 is equal to A plus what? A plus how many d's? 60. Yeah, do you agree with that? Everybody else agree with that? 60's? A plus 6d. And that's gonna be 2 plus 6 times 12. Yeah, what's 6 times 12? David, what's 6 times 12? Oh, 72. 72, so it's 2 plus 72 is 74. So we've decided that the seventh term is 74. Now, you know, we can check that answer by putting in a 2 here, add 12, add 12 again, add 12 again, add 12 again, we're at 50, we're at 50, what'd be the next number? 62. 62, and then add 12 again and get 74. And if you were to find the seventh term in this fashion, that would certainly be okay, but if I were to ask you for a term further out the sequence like A sub 20 or 30 or 40, this might be sort of inconvenient, and this is I think the quickest way then to find out that term. Okay, you have problems like this in the homework now where you'll first of all be given some sequences and you'll be asked, are these arithmetic sequences? Then you'll be given some information about a sequence like in this case, you were given the second term and the fifth term, and you had to predict a term further out. So that's fairly typical of your homework. Okay, now let's switch over to series. If I had an arithmetic sequence and then I added up the individual terms out to say the first in terms, if I added up the first in terms, that would be an example of an arithmetic series. And we have two formulas for adding up arithmetic series. Let's look at the next graphic and we'll see these. You'll find these in your textbook as well. If S sub n is the sum of the first in terms of an arithmetic sequence, now you see, here's the first term, here's the second term, third term, this is the nth term, but the difference now is we're adding these up, that makes it a series. So I'm gonna call S sub n the sum of the first in terms. Then there are two formulas that'll calculate S sub n. One formula says S sub n can be found with this formula n over two times two a plus n minus one times d. But then the other formula, which is a little bit shorter, says S sub n is n times a plus a sub n over two. Now that's the first term plus the last term added together and divided by two. Let's first see where these two formulas come from and then we'll look at how we can decide which of the two formulas we want to apply to certain applications. Okay, if you're coming back to the green screen, let's just look at this problem of the question of where those formulas have come from. So we said that we were going to define S sub n to be a plus a plus d plus, let's see, the next term would be a plus two d. And let's just keep going until I get to the nth term and the nth term is a plus n minus one times d. Just like we were using a moment ago when we were calculating the nth term of a sequence. Okay, now what I'm gonna do next is I'm gonna write that summation down backwards. I'm gonna put a over here and then just before it I'll put a plus d and then if I reverse this order, I'll have the nth term up here at the front. I better move this back just a little bit to give me some more room to fill that in here. That's gonna be a plus n minus one times d and I have all these other terms written in here. Now, if I add on both sides, if I add on the left and if I add on the right, then on the left I have S sub n plus S sub n equals two S sub n. And if I add here, I have a plus a plus n minus one d.  And if I were to add here, you notice this time I have an extra d in the first term but I have one less d in the term just below it. So when I add those together, I'm gonna have the same number of d's that I had over here because I've increased the d's there but I've decreased the d's here because these d's are going down. So when I add these together, again I'm gonna get two a plus n minus one times d. Now let's go to the very last summation down here on the very end. When I add these two together, what will I get? Susan, what will I get there when I add those together? No idea.  Two a plus n minus one d. Exactly, yeah, two a plus n minus one times d. Okay, well the idea is I have exactly the same sum in every one of these columns and this column and the next column and the next column, these all look alike. And the reason is every time I add an extra d, I've deleted a d on the other row. So when I put them together, I still have the same number of d's. So this tells me I have a multiple of two a plus n minus one d's. Now I have as many of these expressions as I had terms listed up here. And remember that was the sum of the first n terms. So I have n times two a plus n minus one d. I have n of them added together, so I'll just multiply by n. So this tells me that s sub n is equal to n over two times two a plus n minus one times d. And this is the first formula that we saw in that previous graphic. If you could, let's get ready to go back to that graphic in just a moment. But there was another formula that was given up there. Let me show you where that one comes from. If I just continue this, I'm gonna separate the two a and write it this way. n over two times a plus a plus n minus one times d. Now where have you seen this before? What is that? That's the last term of the series. Right, that was the last term of the series. This is a sub n. So what we have now is n over two times a plus a sub n. Okay, now the two formulas that we showed in our graphic just a moment ago were the formula given right here, n over two times two a plus n minus one d. And the other formula was n over two times the sum of the first term and the last term. Now, these are both helpful formulas in solving problems, but we can't use them interchangeably all the time. You see, if I don't know the last term, then I couldn't use this formula. But if I were given an arithmetic sequence, I should know the first term, so I know a, and I should know what d is. And therefore, and if I knew how many terms there were in, then I'd put in minus one here. So sometimes I'll want to use this formula, but if I know the first and the last terms, I might want to use that formula. Now let's go back to that last graphic and just look at those two formulas one more time before we go to some examples. Yeah, you see it says a sub n was n over two times two a plus n minus one d. You'll find that formula in your textbook. And then the second formula was s sub n equals n over two times a plus a sub n. Yeah, so okay, now let's go to the next example and we'll see how to apply that. Use one of these two summation formulas to find the sum of four k minus one where k goes from one to 20. Okay, now let me just write that again right below here. We're taking the summation k equals one to 20. Of four k minus one. Now the first question we might ask is this an arithmetic sequence or rather series? Because if it's not arithmetic, I wouldn't be able to use one of those formulas. Well, what would the first term of this summation be when you substitute in a one? Three. It'd be three, yeah, if I plug in a one, four minus one is three. And when I substitute k equals two, David, what would you get for k equals two? Okay, equals two, you'd get a k. Well, if k is two, that'd be seven, yeah. Right. And if I substitute k equals three, Mike, what would I get when k equals three? I'm still working on it. Okay, let's see, four times three minus one, that'd be 11. Yeah, and it looks like what's happening is these numbers are increasing by four. And you notice there's a four coefficient right here. And if I go to the very last term, when I substitute in a 20, that'll be four times 20 minus one, 80 minus one is 70, 79. Okay, so the question is what is this sum? Well, according to my formula, I would have called this S sub 20, the sum of the first 20 terms.  And the sum of the first and last terms, that would be three plus 79, three plus 79. And that's gonna be 10 times 82, which is 820. That's the sum of all these numbers. So without writing them all out and then physically adding them up, I've used a formula to compute the sum to be 820. Now, what if I were to solve this using the other formula? I think we'll still get 820, let's just try that. So in this case, we have S sub 20 and the formula is N over two times two A plus N minus one times D. And how much do we say N was? 20. 20, yeah, that'll be 20 over two times two A. Well, let's see now, A is three, so two A is six. And then N minus one, well, we said N was 20, so that'll be 19 times D. David, what's the value of D in this problem? What's the common difference? What's the common difference up here? Four. It's four, yeah. So we now have 20 over two is 10 and six plus, now four times 19 is, let's see, I think that's 76. So this is 10 times 82 once again and that's 820 just like before. So we get the same answer using either formula in that case. Okay, one more example and then we'll move over to geometric sequences and series. Let's go to the theater problem. It says a movie theater auditorium with stadium seating has 25 rows of seats. If there are 22 seats in the first row and 46 seats in the last row, what is the seating capacity of the auditorium? Now, it doesn't mention this, but this is intended to be an arithmetic sequence. Every row is increasing by the same number of seats. So let's say that here's the first row, then here's the second row, then here's the third row, and then here's the fourth row, et cetera. And we keep on going until we get to the 25th row. So this is row number 25 and this is row number one up here. And on the first row, there are 22 seats. And on the last row, there are 46 seats. That's a difference of 24. So what would you guess is the increment of seats on every row? What does it seem to be the increment in seats here? Each row is increasing by the same number of seats. How much would you say? There are, it's a 24 difference from 22 to 46. And there are 25 rows. I'm thinking that G is probably gonna be, what do you think? One. Is one, yeah, I think D is one. And you might say, but 24, shouldn't it be 25? We'll see, there is no D added on the first row. The Ds only begin on the second row. That's why this number is one less. So if I use a formula, I wanna find S sub 25, the sum of the seats on 25 rows. And we have the first term and the last term, so I think the simplest formula to use here is N over two times A plus A sub 25. And while we're at it, why don't I put a 25 in where the D is? Because that's the value of N, 25. So this is gonna give me 25 over two times, what would be A in this case? 22. Be 22, yeah, that's the number of seats in the first row. And then we're gonna add on the number of seats in the last row, that's 46. And so this is 25 over two times 68, which is 25 times 34, 25 times 34. Now let me ask you, if you had 34 quarters, how much money would you have? 34 quarters, 34 quarters. Well, let's see, if you had 32 quarters, that would be eight dollars. And you'd have two quarters left over. So this would be 800 and two quarters left over 850. So I think there are 850 seats in this theater. Now this problem could have been worked using the other formula too, but in that case you have to use the value of D rather than the value of the last term. Okay, in your homework, you will see problems then on arithmetic sequences and series. And the sort of formulas you need to keep in mind are the formula for the nth term of the sequence and the two summation formulas for the arithmetic series. Okay, we're gonna shift gears now and look at another type of sequence, and this is a geometric sequence. And the distinction between geometric and arithmetic is whereas in arithmetic sequences, the numbers in the sequence increase by the same amount, they have a common difference. In this case, they're gonna increase by a common multiple. Every consecutive term is the same multiple of the one just before. Or you may say the ratio of two consecutive terms is always the same. Let's go to the next graphic and we'll see this. Here's an example of what I call a geometric sequence. A sequence is geometric if the ratio of consecutive terms, a sub n plus one over a sub n is constant. Okay, so the first term again I'll call a, and you notice now we multiply by an r. I'm not adding on D, but I'm multiplying by r, so I get ar, then I multiply by r again and I multiply by r again until I get out to the nth term. Now, if I take the ratio of these two terms, ar over a, you notice that ratio reduces to r. And if I take a ratio of these two terms, I have ar squared over ar, so I get r again. So consecutive terms have a common ratio, and that's why we're using the letter r for the multiplier because it represents the common ratio of that sequence. Okay, and this increases, or this series changes, I should say, geometrically. Now, let's go to the next graphic and we'll see some examples of what are geometric sequences and there may be one in here that isn't. Okay, this example says which of the following are geometric sequences and if it's geometric, find a, find r, and find the nth term. Well, this time what I'm looking for is not a common difference, but a common ratio. So what is the ratio of the first two terms? Mike, what's the? It's five. Yeah, Mike, what's the ratio of the second two terms? And that's also five. And I think the last one's five. So it looks like a is two and the common ratio is five and that becomes the multiplier. Two times five is 10. 10 times five is 50. 50 times five is 250, et cetera. So it looks like in our first example here, a is equal to two and r is equal to five.  The r's were introduced in the second, the third, the fourth term. So I always have one less r than the position number. So using that formula, this'll be two times five to the n minus one power. This is a formula for the nth term. Now, if you wanted to know, for example, what is the 10th term of this geometric sequence? If you wanted to know a sub 10, you'd put a 10 in here and that would be the ninth power. Two times five to the ninth power. It'd be a very big number. You'd probably want to use a calculator to find it, but that's how you would find the 10th term or any other specific term of that sequence. Now, looking at the sequence in B, let's keep that, yeah, there we go. Okay, we start off with one and then we have the square root of two. What's that all about? And then we have a two and then we have a two squared to two. Is there a common ratio? Well, let's see, what's the ratio of these two terms that that would be the square root of two over one is the square root of two. What's the ratio of these two terms? That would be two over the square root of two and can I reduce that? How would you reduce that? Multiply both the top and bottom by root two. Yeah, exactly. You remember if you have a radical in the denominator, the way you reduce it is you multiply by another square root of two on top and you multiply by a square root of two on bottom. And what's the square root of two times the square root of two? Two. Two and those two's cancel and this is the square root of two. Well, things are looking hopeful here. I have the same ratio so far. What's the ratio of these two terms? Square root of two. Square root of two again. So if the pattern that we're seeing is the pattern that's actually intended to produce the next terms, then this is a geometric sequence as well. If you think back to the last episode, you may remember there was an example where it started off one, two, three and I said what was the next number and it seemed like the obvious number was gonna be four but I showed you some formulas that could actually make the next number be anything you wanted. So we can't guarantee that this pattern is gonna continue to be geometric but if we assume it is, it certainly looks like it's geometric from all the information that we're given. Okay, now the questions we were asked here were to find A, find R and find A sub n. Let's see, Susan, what would be A in this case? One. It would be one, okay. And what would be R? Square root of two. Square root of two, exactly. And now who can tell us what is the nth term? The formula is A, R to the n minus one. So that's gonna be? One times the square root of two to the n minus one power. To the n minus one power, okay. And you know, there's really no reason to show the one in front. So we could say just the square root of two to the n minus one power and if we wanna be fancy we could say this is two to the, let's see, that's two to the one half power. And one half times n minus one is n minus one over two power. You could write it that way. This answer is fine right there. This answer's fine and this answer's fine. So you don't have to reduce the exponent but you can see this does look simpler than any other. And I've just changed the square root into a one half power and then multiplied by exponents there. Okay, let's look at the third sequence. We have the third sequence in that same graphic. There you go. Okay, we have one half, one sixth, one 24th, one over 120. Let's see, what's the ratio here? One sixth over one half, one sixth over one half. Well, if I invert and multiply that's one sixth times two over one is a third. That's the ratio of the first two terms. What's the ratio of the next two terms? One over 24. Divided by one over six. Yeah, Mike's already reduced it. What'd you get? A fourth? One over four. Okay, so immediately, Mike, what does this tell you about this sequence? It looks like the next one's gonna be a fifth. Next one's gonna be a fifth but I guess the question is, is this geometric? Is that a geometric sequence? No. No, it's not, because we don't get the same ratio. We're getting a pattern in the ratios, one third, one fourth and I think the next one is one fifth but it's not the same ratio. So it's not geometric and therefore we don't have to find a, we don't have to find r. There's no nth term for us to find but only if it were geometric. Okay, let's go to the example, or the next example. If the first term of a geometric sequence is six and the fourth term is 81 over four, what's the fifth term? This sort of sounds like the problem we were asked about arithmetic sequences a minute ago. We're given two terms and we're asked to find some other term. So let's see, there are five terms mentioned at all so I'm gonna put down five blanks here. The first term is six. Well, that tells us a right away. The fourth term is 81 over four and we wanna find out what is the fifth term. What is the fifth term? Well, let's see, we have a is equal to six. We have a sub four is equal to 81 over four. Now, you know there's a formula for the fourth term and that would be a times r to the, what power should I put on that r for the fourth term? Three. Third power, yeah. You remember for a sub n, I put n minus one r's. So for a sub four, I put three r's. So that's gonna be six r cubed. Okay, well, let's pull that out and solve it. We have six r cubed equals 81 over four. That says that r cubed is equal to 81 over six times four. Now, you know, we could cancel a three there and that's gonna say r cubed equals 27 over what will the denominator be?  Eight.  And so r will be the cube root of that, which is three halves, r is three halves. Okay, we have a, we have r. So let's find a sub five. Well, you know, probably the quickest way to find a sub five is just multiply the fourth term by three halves. So the fourth term, let's 81 over four. And if I multiply it by three halves, I'm gonna get 243 over eight. So it looks like this is 243 over eight. But the other way to find that term, just in case I weren't asked the next consecutive term, where it wouldn't be quite so convenient to do that, to find the fifth term, I could solve it this way. A sub five is a r to the fourth power, which is six times three halves to the fourth power. Which is six times, let's see now, you know, three squared is nine, what's three to the fourth? If three, this is three squared times three squared, three to the fourth. 81? Yeah, yeah, what I'm thinking of is this is, when I say three to the fourth, that's three squared times three squared, that's nine times nine. So nine times nine is 81. So that numerator is 81. And two to the fourth, well, that's two squared times two squared. How much will that be? 16? 16, right, very good. Now, if I cancel off a two, I have three times 81 over eight. And that's gonna give me two, 43 over eight. And that's exactly what we got up here. Now, clearly this is the longer way to get to the answer. If you know the fourth term and you wanna get the fifth term, just multiply by r. But if you had been asked to find some term further down the line, this might be quicker than finding out all the terms in between. Okay, so the question was, what's the fifth term and it's 243 over eight? Okay, let's go to the next example. Oh, actually it's two formulas. Now, we come to the question of how do you add up geometric series? Well, the two formulas that we wanna use, first one says, if you add up the first n terms of a geometric series, that would be a plus a times r plus a times r squared up to, here's the nth term, which can be abbreviated this way, then that summation will be a times one minus r to the n over one minus r. Now, we wanna verify that formula here in just a minute, but this is how you add up the first n terms of a geometric series. You know, the summation here goes from one to n, and this is a, r to the k minus one power, because in the kth term, you have one less r than you have the position. That's how all of these have been going. Now, there's a separate formula when you have an infinite series. If the absolute value of r is less than one, that is if r is somewhere between negative one and one, and you wanna add up an infinite geometric series, so you notice this just keeps on going, there's a dot, dot, dot over here. This goes from one to infinity, so k goes from one to infinity, a, k to the k minus, a, r to the k minus one power. Then we have this formula for the summation of the series, a over one minus r. Now, let's just see where those two formulas come from, and then we'll apply them to some applications. Okay, so we're talking about adding up, first of all, the first n terms of a geometric series, that's a plus a, r, plus a, r squared, and when we come out to the nth term, we said the nth term was a, r to the n minus one, but there's a formula that we were given on that previous graphic, let's see where the formula comes from. What I'm gonna do is multiply both sides here by r, multiply by another r, and so I'll multiply each of these terms by r. I'll get a, r, after multiplying this guy by r, and then for the next one, I'll get a, r squared, and if I keep going, you notice what happens is all these terms, all these terms just get pushed over by one and their exponent goes up one, so I'll have an a, r to the n minus one here, plus a, r to the n, that's where I multiply the very last term by r, so I get a, r to the n. Now, if I subtract these two, if I subtract these two, I have s sub n minus r, s sub n, and when I subtract over here, I get a lot of cancellation, these cancel, these cancel, everything in the middle cancels, so what I have left is a minus a, r to the n. If I factor out the s sub n, in parentheses, I have one minus r, and over here, if I factor out an a, I have one minus r to the n. So this tells me that s sub n is a times one minus r to the n over one minus r. So if I know the initial term of a geometric, of a finite geometric series, and if I know the number of terms in the series, then I can calculate the sum of the terms in the series by using this formula right here. Let's just take an example of this before we look at the other formula. Let me just write that formula here at the top of the green screen, and I'll show you an example of how this would work.  Now I think this is a geometric series. What would you say is a in that series? One. One, and what's the multiplier in each series? Two. Two, yeah, or you could say, what's the common ratio of consecutive numbers? Two over one, four over two, eight over four. It looks like r is two. So what is the sum of that series? Well, that sum would be the sum of the first five terms, s of five. And we said we have a formula for that, and it looked like a times one minus r to the n over one minus r. Well, a is one, r is two, and n is five, so that's one plus one minus two to the fifth over one minus two. And that'll be one minus 32 over negative one, which is minus 31 over minus one, which is 31. 31 should be the sum of these numbers right here. We get 31. Now, of course, we're using a formula to get this answer. Let's try adding them up a different way. Look, here's a 10 right there, and here's a 20 right there, and then there's one more. So I've got 10 and 20 and one, sure enough, that makes 31. Now, you might say, well, Dennis, you've just managed to make a mountain out of a mole hill. It was really easy to add those up. Why did you go through all this trouble to find the sum of those numbers? Well, of course, if there are only five numbers in the series, then it's not difficult to add those, but if there were many more numbers in the series, this formula would allow me to calculate that sum anyway. Okay, can we go back to that last graphic with the summation formulas on it? I just want to look at the other formula that's on there. Here we go. You remember we said that if I'm adding up an infinite, if I'm adding up an infinite geometric series, that the formula is s equals a over one minus r. Okay, now let's just see where that one comes from. If you come back to the green screen, this said, we said that s sub n was a times one minus r to the n over one minus r. Now, let's assume that r is between negative one and one. Or the way it was written in the graphic, the absolute value of r is less than one. Now, in other words, this number might be a half or a third or negative a fourth. What happens when you take a number between negative one and one and raise it to larger and larger and larger powers? What's going to happen to that? You get smaller and smaller. Yeah, this is going to approach zero. So when I look at s sub, maybe I should say infinity or on the graphic, I think I just called it s, as the number of terms becomes infinite, this nth power goes to zero, and so I have a times one minus zero over one minus r, which is a over one minus r. That's the formula for adding up an infinite geometric series, but you can only apply it in the case where r is between negative one and one. Okay, let's go to the examples and look at how we could apply those last two formulas. In the first example, it says we're going to find the sum of one minus a half plus a fourth minus an eighth plus, it's a dot, dot, dot, and this is minus one half to the n minus one power. You see, it looks like every time we're multiplying by negative a half, and that causes the signs to alternate. So I put a negative in my multiplier right here. A is one, so you don't see a in front. R is negative a half, and we raise it to the n minus one power. Now what's the sum of this series gonna be? Well, we have a formula for this, and that is that we're supposed to put a times one minus r to the nth power over one minus r. Well, a is one, we've got that over here, a is one, so I can forget that term. R is negative a half, and I'll put negative a half right here. So if I just replace those, if you come to the green screen, I'll have room to write that out. If I replace those with the number of values, we said that a was gonna be one, r was gonna be negative one half, and we wanna calculate the sum of the first n terms, and that's a times one minus r to the n over one minus r. That's one minus negative a half, and I'll substitute for r right here. That was a negative one half to the nth power, and so a was one, so we have one minus negative one half to the nth power over three halves, one plus a half, and if I invert and multiply, this is two times one minus negative one half to the nth power all over three, and that's the sum of that, that's a formula for the sum of that first finite series that we were given in that example. Okay, let's go back to that graphic and look at the second example. You notice in the second example here, we have an infinite series, because this just keeps on going. There's a different formula for this, and the formula is that we should use s equals a over one minus r for this infinite series, and I can use this formula if r has absolute value less than one. Well, let's see, what is a in this problem? What is a? One fifth? No, not a, a is the first term. So a would be two, yeah. Now, what are we multiplying by every time to get the next term? One fifth. Yeah, that's where the one fifth comes in. So one fifth has absolute value less than one, and therefore the sum of this series is gonna be a over one minus one fifth. Now, I believe if you reduce that, I don't have much room here, but I think you're gonna get five halves. That's the sum of this infinite series. Okay, now, let's look at this last example, and this isn't actually a series, this is a decimal number. Now, you're probably familiar with a decimal, if you put a bar over digits, what that means is those digits keep repeating, so this is 0.81, 81, 81, 81, because the 81 keeps repeating. Let's come to the green screen, and I'll show you how we'll convert this to an infinite series and figure out what rational number this is. We have 0.81 with a bar over it. Now, that means 0.81, 81, 81, and that goes on forever. Now, I can write this as a series if I say 0.81 plus 0.0081 plus 0.00081, and you see what that does is it just keeps shifting the 81s over a little bit. It looks like a is gonna be 0.81, and it looks like the multiplier here is 100th. Every time I'm just moving the decimal back two places or forward two places, so that means I'm just multiplying by a power of 10, and this is an infinite series, and so 0.81 with a bar over it must be, that's the sum, must be a over one minus r, which is 0.81 over one minus 0.01, which is 0.81 over 0.99. Okay, so now if we just reduce that, we'll know what is the rational number that is this decimal. That's 81 over 99, and if I reduce that, if I divide by nine, I get nine over 11, nine over 11. So this repeating decimal is the same thing as the fraction nine over 11, and I found it by separating this into individual pieces and noticing that every time I had a common multiplier to get to the next term. Okay, let's go to the next graphic, and this problem ought to get your attention, I think. Here's a way that you can become actually quite rich. It says, congratulations, you've been offered a job that pays one cent on the first day, two cents on the second day, four cents on the third day, et cetera, so that your pay continues to double every day thereafter. What is the total amount you would be paid for 30 days? Now, let's go to an interview that I had with some students out in front of the library where I asked them a question very much like this. Well, let's see, class, we're just outside the library, and I have a really interesting question for you here. Which one would you choose? So you'd be thinking about this. So let me ask a student here what she would do. Tell us your name. My name is Yurango. Yurango, and what is your major, Yurango? Drafting technology. Drafting technology. Well, you could possibly be very rich with this plan. What would you choose, $100 a day for 30 days, or one cents, two cents, four cents? I think I'll choose second. You'd choose the second one, why did you choose that? It sounds so small. No, it's not small. Oh, it's not small, okay. On a fourth day, it will be like 16 cents. 16 cents and then 32, yeah, it starts to grow very fast. You know, actually, you've chosen the one that would pay you more money, but let me change it. What if I were to give you $1,000 a day every day for 30 days, let's see, for 30 days, it'd be $30,000. Would you still choose this, or would you rather go with a thousand? A thousand. You'll go with $1,000 a day? It's 30 days. 30 days, $30,000, that's a lot. Billy, what's your major? I'm majoring in biology. In biology, and I think you just told me a minute ago that you're in college algebra right now. I am taking college algebra. Yeah, so in other words, this is in his course, although I bet he hasn't gotten to it yet in his course because we're filming this a little bit earlier than the actual studio lecture. So Billy, here's the choice, $30 a day, whether $100 a day for 30 days, or one cent, two cents, four cents, et cetera. What would you choose? Well, I think definitely I would choose this one. Why is that? Well, it multiplies very fast. It, I don't know, expounds exponentially. It goes up exponentially, good words, okay? So by the time you get towards the end of the days, you're making a great deal more than $100 a day. Yeah, yeah, it starts off pretty low, but it does get to be bigger. So I'll tell you what, what if we changed it to $1,000 a day? $1,000 a day, it's pretty close. What, $30,000? Well, I think I would have to add up the totals and for 30 days, $30,000. And then- What's your gut instinct on that? I would probably just go with this because it multiplies a lot quicker. Yeah, yeah. And anytime that it goes to an exponent, it's gonna go a lot faster. Yeah, okay, Billy, here's my last offer. All right. What if I gave you $10,000 a day for 30 days? What's that, $300,000? Where can I get a job like that? Pay me $10,000 a day, sounds good. Would you rather do that? Just off my gut instinct, I'd probably do this just because you might be earning millions, who knows? As a matter of fact, Billy, this, you'll make about $10 million here. So hey, thanks a lot. All right. Very good. Thanks. Okay, let's see why that would be over $10 million. If we go to the green screen, let's say on the first day, you'll be paid one cent. So I'll write that as a decimal. On the second day, you'll be paid twice that much, two cents. And then four cents, et cetera. And we're gonna go out here until we get to the, this is the 30th day right here. I won't bother figuring out how much that is. But I wanna calculate the sum of this finite geometric series. You see, it is geometric because A is one cent and R is equal to two because we're doubling every time. And to find S sub 30, our formula is to say A times one minus R to the N over one minus R. And I'll be substituting in 30 for N. I'll put in two for R and I'll put in 0.01 here for A. So that'll be 0.01 times one minus two to the 30th power over one minus, one minus two. And that's gonna be 0.01 times. Now let's see, you know, this is gonna be a negative one right here, so why don't I just reverse the order up on top and say two to the 30th minus one. Now I'm just about out of time, but I tell you what, I'm gonna leave this to your devices, to your calculator, you and your calculator to find out that this is more than $10 million. You'll just have to raise two to the 30th power, subtract one, and then multiply by 0.01 because you were getting one cent on the very first day. Okay, I think we're out of time. I'll see you next time for episode 33.


We're having the genetic code changes the way we look at everything. Really allows the individual and individual creativity and insight and intuition to really come to the fore again in biology. In 1960, there were so many things we said, will we ever be able to do these things? I don't think we never even thought about sequencing the genome. Such simpler things seemed daunting at the time. And here I was sitting 30 years later and there was a sequence of the genome in front of me. And I just had this sense of history coming together, the pace of history, moving at an incredible rate. And in fact, it has changed biology, there's just no question that we've moved into a new era. We're accumulating data in orders of magnitude more rapidly than we were before. In fact, I would go so far as to say, for the first time we now have the tools commensurate to the task of understanding biological systems. There's going to be a bright line in history and the bright line in history is going to start with the first genome going forward because it's so revolutionary, completely revolutionized as how science is done, the pace of which science is done, the pace of which discoveries are made and will ultimately revolutionize medicine. Most of the analysis of genome sequences is really focused on the genes for obvious reasons, but the bits of code for proteins and how many proteins they make and what of those proteins are, what the so-called proteome of each individual genome is. What that has revealed is not simply where the genes are, which is an important thing to know, but it's shown us that mother nature is conserving far more than the genes. It's conserving segments up and down the genome, and for the most part we don't have the slightest idea why she has done that. This is going to be grist for the male for the next 10 to 20 years. For protein coding reasons, we have the genetic code, we've had that for some time. We don't really understand how to interpret the DNA sequence of the rest of the genome, and the best method, it turns out, for doing that is to rely on evolution. Let me take the example of the fruit fly. If I showed you a Drosophila melanogaster and a Drosophila pseudobscuro, for all intents and purposes they would look identical to you, yet they're 60 million years different in evolution. If you look at the genome sequences, only those sequences that had some selective advantage that were important would be conserved. I think a lot of the challenges in genomics are really to understand what the rest of the DNA does in the organism. Comparative genomics, I think, is going to be one of the most exciting areas for the next probably 10 to 20 years. For one thing, I think it is the only way we're going to really begin to understand the human genome. For myself as a mouse biologist, one of the most exciting things to do right now is to take a segment of a chromosome from mouse and compare it base for base to the same segment from the human genome. For one thing what it does is it tells us it identifies every single important segment of that chromosome because it's the part that mother nature has chosen to conserve in the 50 to 75 million years since mouse and human separated from one another. The 21st century is rapidly turning into something that's half experimental and half an information science, a science where there's as much to be gained from mining data in fast public databases. You will not be able to do biology going forward without being incredibly computer literate. We'll be able to eliminate 99% of the random experiments and use the computers to narrow down using all this information of what is the key experiment to do that will advance science. Now the scientists mid-century, they purified the DNA molecule away from the cell. The human genome project now purified the information away from the DNA molecule and that's going to be as profound transformation in this century. Having the genome sequences there, having it available on a computer really allows the individual and individual creativity and insight and intuition to really come to the fore again in biology. Now when everything's in the database, people can begin to think and design particular experiments. They can go back to the original idea, the creative idea and they can be much more enabled. So I think it really enables the small research scientists. It also enables the theoretical biologists. So if you had asked me five years ago to name the successful theoretical biologist in the world, I would have named Francis Crick and there would have been a long pause and I'd say I'm sure there must be somebody else but I can't think of who they are. I think now we're really engendering a field where these massive data sets are being generated, made freely available. There's really an opportunity for people to come in and interpret data. It really brings the intellectual content back into research and really makes the individual, not even the individual group but the individual person a real unit of scientific discovery again. So I think it's tremendously liberating. I think this is the groundwork for the human biology and the medicine of the future. As a physician it is the medical implications of the genome project that have always seemed to me the most compelling. The reason to do this, the chance to alleviate suffering, to cure diseases that we currently have little to offer for, that's why we're doing this. My favorite way of looking at the impact of the genome is a story which actually didn't involve sequencing of the genome, involved sequencing of a gene. It was a gene that we discovered years ago in a cancer inducing virus which controlled the growth properties of cells, was an oncogene, and was discovered to be in a human cancer, the critical element that led to the human cancer. And Novartis, drug company, took that information and built a compound which inhibits that enzyme. And that has now revolutionized the treatment of the disease, chronic myelogenous leukemia, the drug being Gleavec. That's the poster child, because that's what basic molecular information can lead you to. It can lead you to drugs that are specific, that are powerful, that are safe. So we can design preventive drugs or design proteins or design genes or whatever. And in doing so, I expect we'll add easily 20 plus years to the average lifespan, and those 20 plus years will be productive and creative years. And parenthetically, that poses a really interesting challenge for society of how we treat older people. Well, knowledge, particularly large increases in knowledge, always have major effects on society. And this is knowledge about living organisms, and of course that includes us. While there's a genetic component to almost everything, there's a huge difference between the physical and the actual life outcomes, and the future of medicine will have to deal with that difference. And this quite naturally leads to the whole issue of genetic privacy. So if we know all of your predispositions, who has a right to have that information, does your insurance company, does your employer, does your family, do your friends? I mean, where do we draw the boundaries for these kinds of things? A lot of patients are going to push their doctors for information, which is going to force the doctors to educate themselves, even if they don't want to necessarily.  But patients will, I think, drive the medical community even faster than they were prepared to go by asking the right questions and not resting until they get the answers. We're going to see miracles. We're going to see miracles of all sorts coming out of the molecular information that we have, translated through chemistry, translated through clinical experience, and that's what this is all about.


Hi, I'm Luke, I'm the Helmet Buyer at Ellis Briggan Mountain Sports. I'm doing this video to give you a bit of an insight into helmets and why there's so many various prices and the differences between them all. Gone are the days where helmets are huge bulky things that look like you're getting sent into space. The new modern day ones are a lot lower profile, lighter and more comfortable to wear. The main thing to look at in helmets is the different types of construction. There's two main types. We have in-mold helmets and hard shell. The way in-mold helmets are made is using a tough polycarbonate outer and fusing it in one moulding process to the EPS energy absorbing impact liner. As a result you tend to have lighter helmets that are thinner and often with more complex venting systems that you can open and close. Hard shell helmets are made slightly differently. They have a tough wearing outer and then an energy absorbing foam inside. They're made in two separate parts which are then stuck together or attached together. As a result you tend to end up with a slightly chunkier helmet which are great for freestyle uses that are a little bit harder wearing. But you tend not to be able to do much with the ventins so the ventins are usually permanent full-time ones, not ones that you can open and close. But the way they make them also keeps the cost down. You may also come across hybrid construction. This is a mixture of in-mold and hard shell. You tend to use in-molds to keep the weight down and then the hard shell in the areas which tend to get more knocks. Downside is sometimes they're a little heavier and tend to be more expensive. Another safety feature you'll come across in a helmet is MIPS. This stands for Multidirectional Impact Protection System and is something that many different companies use. The easiest thing to look for on a helmet is a little yellow sticker or stamp on it which will indicate that it uses MIPS. Ventin a helmet is really important because it allows you to regulate your temperature on the mountain. There's two main types. There's what we call passive and active. Passive ventin you usually find on cheaper helmets tends to be full-time holes within the lid that you can't open or close. It does keep the cost down though. You'll often find it on park helmets too. Active ventin you usually find within in-mold helmets usually but not always. It's a more complex system that will allow you to open and close the vents and alter how the air flows throughout. Sizing on helmets is really important because if it doesn't fit your head properly it won't work as effectively. As a result there's different fit systems. There's what we call standard so the helmets mark the standard that is non-adjustable. The shell would just come in various sizes to fit different head sizes. There's auto-adjust which usually uses some form of elastic band at the back to fit on your head and just adjust slightly. There's fully adjustable. If the helmet is fully adjustable it will usually have some sort of dial or thing at the back of the helmet which can be tightened or loosened for different head shapes and sizes. It doesn't always pull just in at the back of the sides. It usually pulls in around the helmet as a whole to give a better, more secure fit. Snow Sports helmets have to pass various vigorous tests to be classified as such. These are different to motorbike helmets or cycle helmets because of the conditions you're facing on a mountain as opposed to on the roads. If ever you're unsure of whether a helmet is suitable for Snow Sports you can always look under the liner and there'll be a sticker in there somewhere which either has ASTM 2040 which means it reaches the US safety standard or EN 1077 which is the European safety standard. Year on year helmets are getting better. Brands are bringing in their own additional safety features. At the end of the day you do kind of get what you pay for with a helmet. A cheap helmet would tend to have more basic vent in and more basic fit system and construction where the more you pay the better the fit system would be and the better the venting control. If ever you're unsure on a helmet either hit us up at the comments below or pop into store it's really important that you make sure your helmet fits well and we can do that for you by measuring up and trying on some different models.


it always surprises me how many screenshots I actually take. I take a look at my screenshot folder and I have screenshots going back over the last four years since I became an organized person and I just have hundreds and hundreds of screenshots.  Like I go back through these folders and I just like, why did I take a screenshot of this? It doesn't make any sense. But the point is, is that I take a lot of screenshots and in order for me to be happy with the process of taking a screenshot, I've been searching for an application that does screenshots really well. Now there's a terminal based application called Scrot and it works reasonably well. It has a few features, but it's mostly a minimal screenshotting tool. It just does what it's supposed to do. It takes a screenshot, but I needed more functionality than that. I wanted to be able to go through and annotate things. I wanted to go through and have more control over what portion of the screen is actually being captured. And while you can do some things like that in Scrot, you can't do a lot of things in Scrot that I actually wanted to go through and do. So a few months back, I found an application called Flameshot. And now Flameshot is an application for screenshotting. That's pretty much all it does. However, once you've used the application, you'll know that it does a ton more. So that's what we're going to be looking at today. The app is called Flameshot. And this is what it does. So now let's go ahead and jump over into my desktop. And what we're going to do is take a screenshot. Now you could press the print screen button if you wanted to do that. And that would work on several desktop environments that would probably use the built in screenshotting tool. But my problem with functionality like that is that I have dual monitors. So a lot of times what that print screen key will do is it'll take a picture of both monitors, it will capture both monitors, and then you get this long elongated image. And it's not all that useful in order for anybody to actually see anything that the zoom weigh in, it's not a great experience. So Flameshot, which I have bound to the key binding control all an S looks like this. Now, we'll talk about these key bindings here in a second. But let's just say I want to take a screenshot of this part here. Now, as you can see, it doesn't take a screenshot automatically. It goes through and actually just highlights what you're going to be taking a capture of. And then you can actually move this around as you want before you do anything else, you can move it around, make sure that it's precisely size and stuff like that. And then the cool thing about it is that there's just a ton of stuff that you can do. So you can do you can draw squares, you can draw circles, you can go through and draw arrows. You can do text. And you can go through and blur things if you wanted to go through and blur things if you wanted to. There's just a ton of stuff you can do and you can actually upload it to image or if you want right from here, if you want.  And then if we open it up, this is what it looks like. And that's the screenshot that we took. That is really cool, right? It would go through and allow you to create a screenshot that is perfectly sized and can be altered. So a lot of times with the screenshot tools that you can use to actually select a portion of the screen. Once you once you let go of the selection tool, it takes the screenshot. Now there are ways of delaying that's like if you use spectacle or use GNOME screenshot tool, you can add in a delay if you want to, but it doesn't actually stay there. You can actually keep flame shot open up as long as you want. You can go through and this window here stays open as long as you needed to stay open as long as you're going through and doing stuff. You can just keep that open doesn't go through and close until you've either saved it or closed it. So there are there are a few other things that you should know, along with these key bindings. So basically what this does is that you can go through and show the color picker picker if you want to do that. And that would actually change the color of certain aspects of the application you can go through and use the mouse wheel to change the tool size. And when it says the tool size, it means things that are inside this. So for example, if you use the scroll wheel while you're in here, you can actually see that it kind of changes the cursor a little bit. And it basically what that does is it changes like the stroke around the circle, right? And that's what the scroll wheel does. Those other key bindings that you saw on that screen control c control would copy this screenshot right here automatically to the clipboard. And control s just saves the screenshot to a file somewhere. Usually remembers the last place you saved it. Now the biggest problem I have with flame shot is that depending on what size screenshot you're taking, these buttons move around. And that's not a great thing. So for example, if you have a really small screenshot, you see all these buttons are kind of in a willy nilly place. And the larger you have, they're kind of spread out a little bit more. And the biggest problem there is that they move around. So if you don't use the key binding, and you want to find the save button, the save button isn't always in the same place. Now it's over here. Now it's over here. Now it's down here and so on and so forth. That's just kind of the way flame shot works. I highly recommend going through and you learning the key bindings such as they are, they're not a lot of them. Just control seeing control as are really the only ones that you absolutely need to know. And then you can go through and avoid having to hunt out those buttons which move around. So that is flame shot. Now actually, before I move on what I should do is show you how it starts so you can actually go through and do this from the terminal. So you do flame shot GUI would go through and open up this tool right here. And the reason why this is important is because you take this and you put it into your configuration file for whatever window manager you're in or whatever desktop environment you're in. And that way you can assign a key binding so that when you do a key binding, it comes up. That's what you need to know. Alternatively, you can actually just go through and open up flame shot like that. And it will go through and put a icon up here in your bar if you have a taskbar enabled. And you click on that and then this comes up as well. So personally, I prefer using a key binding. But if you want to go through and use that little icon up there at the top, you can do that as well. I'm also sure you can go through and bind it to the print screen option if that's something that you wanted to do. So that is flame shot. It's really cool. And it's just something that I use multiple times a day to go through and just take a screenshot. It's a really useful tool. And I like it best because it's distro agnostic. I just go through and install on any distro that I'm on.  And it just works. I don't have to worry about whether or not I have it installed because I'm just going to go install it anyways. Like it's something that I explicitly have on my system. And that means that I know that it's there. I don't have to worry about trying to figure out what the screenshot tool is called. So like if you're on GNOME, you use GNOME screenshot. Not that big a deal. If you were on KDE, it gets more confusing because their screenshot tool is called spectacle, which is not intuitive at all. Unless you actually know that's what it's called. Going through and trying to find that can be kind of a pan yes. So again, sorry about the dog. She decides to bark every time I do a video. So that's just the way things go. So that is it for this video. If you have comments, you can leave those in the comment section below. You can follow me on Twitter at Linuxcast. You can support me on Patreon at patreon.com slash linuxcast. Before I go, I'd like to take a moment to thank my current patrons. Today, Devon, Patrick, Elm, Marcus, Megalyn, Jackson, Neff, and Tool, Steve A, Sebrega, Linux, Garrick, Mitchell, Arsene, Carbon Data, Jeremy, Sean, Odin, Martin, Andy, Merrick, Camp, Joshua, Lee, Day Dog, Peter, A, Crucible, Dark Minute Six, Vlad A, and Primus. Thanks everybody for watching. I'll see you next time.


Thank you for coming up here and sharing your incredible testimony. Can you tell us your name and where you are from? So my name is Nellie. I'm from Cleveland, Ohio. Okay, Nellie. And tell us what why did you come here last year in the summer 2018 internship? So last summer before I came here, I kind of was experiencing like like a stagnant like relationship with God. Like I wasn't going anywhere. I wasn't doing anything.  Like I kind of want like a touch from you and like little did I know God had way more in store. Come on. Now tell us when you came here to the internship, there was a problem that you were facing internally. Can you tell us a little bit of when it started, what was happening with your life and the freedom that you received through the internship? So for about three years, I was being oppressed by a spirit of depression.  I was a good student, but I'd come home and I would sleep all day. I'd watch movies. I was like a couch potato. I wouldn't do anything. And I would have like a really hard time sleeping at night. It got to the point where I'd get like one hour of sleep or it would take me three hours to fall asleep. And it was it was just like it was like I was being tormented at night. And the reason why this depression was like afflicting me was because I had a best friend. I grew up with this best friend. Like she was so dear to my heart. But there was a situation where she betrayed me and I was I was so hurt. And it was like how could somebody who loves me so much hurt me so much? And so it kind of just got to the point where I like I literally just fall off the map. I would stop talking to people. I would isolate myself. I go to school and I just had this like expression. Like it was so sad. Like I didn't care. I didn't want to do anything. Like I just I didn't feel like living my best life. Wow. Now tell us when did you realize that you were actually having a depression problem? When did you recognize? Wow, this is a problem. What happened? So we were in class during the summer internship and we actually had a lesson we had like a week-long lesson about deliverance and so we were naming off the like symptoms and characteristics and I was able to check off each one of them and at that point I was like God, I have a problem.  How did you receive your freedom? And what does it look like now? So we had a prayer line and we was getting prayed for and like during the prayer line, I received my freedom from that spirit of depression. Yes. Amen. Come on. And so my old life it looked like it was sad. It was it didn't even look like it was worth living. But now it's like it's like I have a new identity. I'm so happy. Like it's like I have a choice to be happy now. Like sadness isn't my identity. Come on. And how do you sleep now? I sleep so good right now. Like it takes me like I fall asleep like that. So I feel great. You guys, I'm a personal witness to this is that when she was here in 2018 you could tell her countenance just by looking at her that there was this grieving sadness about her and I'm telling you that when she came back for the second round and I see her just Smiling and just like just full of bliss and I said I'm like there is you are definitely different And I am a witness and many of you guys are witnesses to that too How God has truly impacted her and has set her free and she's here for the second round now Tell us why are you here again? So actually since last summer like I had this desire like it was like Like God put it there. I just had this desire to come again and just like devote more time to God And Honestly, you can never go wrong with investing more time into God Yes, amen, that's right.  Come on. Amen That is powerful you guys that is powerful that is true freedom that we can experience with Christ now Tell us what would be your word of advice for someone that wants to sign up for internship or you know It's contemplating to encourage maybe their children to come on to the internship What would be your word of encouragement for them? Honestly, if you're debating between going or not going go for it Like take that risk like living that Christian life is all about risks like just do it Like invest more time in God like just do it. Come on. Amen. Thank you so much Beautiful testimony. Amen. Were you guys blessed by that? Amen It's it's also to recognize that maybe you were living a basic life where she just went in and out of her day Where she just saw that it was just a sadness not realizing what that that spirit actually took over Where she was causing her to be isolated where it caused her to have sadness and grieveness and not enjoy the life That God has given us right to live that abundant life and the truth was right in front of her that veil was torn off And she said wow, I want to live on the other side. I want to live a life of joy I want to live a life that God has for me and now she is and like I said, I'm a true witness to that seeing her It's literally day and night difference where she's completely happy and full of joy. It's a beautiful thing. Amen


I said, bro, when I came on, I had no idea. Like, Faizan. Oh, I had no idea. Cause you Wayne is huge. Yeah, but Faizan. Were you watching him before? You actually, I had seen him before. We ain't thinkin' nothin' about it. I'm just like, oh, he doin' the internet stuff. Internet stuff. Because we old. Yeah, yeah.  Faizan, Faizan. Yeah, we on boss talk one on one, one on one. Yeah, we gon' talk. He was like, yeah, we doin' this skit, and in my mind I'm like, I'm an actor. I don't know if I should do this skit. That actor voice, yeah. I'm gonna act right up this year, let's go. I'm gonna be dumbin' myself down for this thing. Cause some people don't even know who you think that skits are below them. No, no, no, no. I'm not even an actor. No, no, no, no, no, no. Most people. Most actors. Yeah, cause they was like, Mike, what are you doing?  I said, bro, when I came on, I had no idea, like Faizan. Oh, we gonna get out. I had no idea, cause you're wearing this, huge. Yeah, but Faizan. When you watching him before, you actually, I had seen him before. Well, we ain't thinkin' nothin' about it. I was like, oh, he doin' the internet stuff. Internet stuff. Because we old. Yeah, yeah.  I'm old. Faizan, we asked that question.  But what about the guys like, you do have different people that's finding new ways and new lanes, like the country, Wayne, and all those guys doin' they thing. Is that the same, do you guys have the same respect?   I don't, on Facebook and Instagram, videos and all that, that doesn't pay money. Yeah, he makes some good money. But see, I said, no, but as I asked him the dinosaur, hey, how do planes makin' money? You talkin' about the big old metal things behind the sky?  So I understand, my mind was like, it skits, and I told Wayne, I'ma do it to get some followers. So I'm like, you know, I'ma get some more followers on my page so I can get my verification, you know. At that time, how many followers did you have? I had 111,000 when I met Wayne. So I do the first skit with him, I was supposed to come and be Blake's daddy, and then I was supposed to do part three or four and be gone, came. I remember this episode when you came with Blake, I was like, oh. I was in and out. Yeah, yeah, yeah, yeah. Man, me, man, not just Wayne, but, cause his team is his family, his brothers, Harvey and Tay, his kids, and then Ro. It's a family, and I just fit right in. I just fit, and I felt like, like I knew these people forever, you know what I'm saying? And then Wayne, I don't meet dudes and be like, man, this is like my brother, man. I really love that dude. Like he is a, let me tell you something. That genuine spirit, that's what it is. You ain't gonna meet no celebrity at that point. This is the thing. This is why people don't think he's so big, like Faizana, because of how he care yourself. He ain't walking around with 1100 chains on the net. He taking care of 700 people. Why take care of his whole family? And he's still living like a king. I went to that nigga's house, I say, nigga, what else you do? I thought nigga was still selling dope. And that wasn't a new house that he just bought. No, he just got a new house. Right. He got a million dollars in furniture, I bought that new house.  And I was like, all right, cool. I'll do it for a time. Because first of all, it was just a chemistry with him. And it was fun, you know. I went to that nigga's house. I was like, okay, what else do you do? Wow. And he broke it. So as I got closer to him, he started, because he want everybody to win. He like, bro, you got 270,000 followers. You ain't making no money.  I'm like, I can make money. He showed you.  There are so many different avenues nowadays compared to what it was. Now, Phazon spoke on the residuals that he make. I know what residuals get. Okay. I get residuals.  No, no, no. Yeah. You just get paid for what I get paid for now. From 92. You see what I'm saying? Yeah, yeah, yeah, yeah. So I don't know how that's gonna work if when they start going, hey, dig it, do whatever he does. I'm like, what does he do? Let me tell you something about residuals. He's gonna say he got residuals. Listen, man. Listen, man. Didn't he say that? Because he's been doing work for a long time. Ever since Friday. Okay, let's talk about residuals. Let's talk about residuals. Compared to what you're doing now.  Unless you're friends. Y'all know residuals, I get residuals. I'm a professional actor. So yeah, okay, I made, I did raising Deion. I did about 125,000 off of that, something like that. When the residuals come in, they get smaller. Yeah, it's time to go longer. That's the reality. That's the real, that's any actor. It go all the way down to, you get a check for six cents. I've got a check for six cents before.  That's a waste of people.   I'm an actor. We a lot of people tell them that because we don't wanna let people know. That I'm a comedian. I'm gonna keep it real. But he says, his friends on the Forbes list that do comedy. He's the main guy doing that, but it's a lot of other ones, you know, that's kind of following that whole trend. So that's a whole new thing. They keep saying it, but I haven't seen nobody rich from it. He made a couple of million from it. I don't think he said it, but I was like, people I know that do this are in Forbes. Yeah. So it's like, and there's a track record, they're like, okay, no, this is a real check. It was a real check. Not like the, we heard check. Okay, let's talk about numbers. Yeah. We ain't gotta talk about Googling before. Let me tell you, let's talk numbers. Okay. The skits. Let's say you got, you have a, let's say you got a hair, hair business and you got a clothing business, right? Okay. And y'all, how do TV, how do you make money on TV? How does TV show to make money? Is it because of the people that's picked up and they, the network paid? Ads. Yeah. Commercials and ads is how you make money. I get it. Okay, boom. You got your hair, you got your clothing. We a, are you ready? We gonna make a move. We gotta invest in some ads. Snowfall is the number one show on FX. Right now. Great show. Five million views a week. Five, that's a lot. Right. They got 10 episodes. Let's get the calculator out. So five million. Time 10. Times 10 is 50 million views. Now hold on, y'all ain't made our decision yet. Y'all still shopping around. Now country Wayne does a million views a day. So that's seven million views a week. But the difference is he ain't got 10 episodes. He got 52 weeks. So times 52. That's 364 million views.  I'm gonna put my ads on country Wayne. Exactly. So that's what I'm saying. Times is different. Times and times. So what happens is when you, and I understand what phase lines come from because phase lines are alpha male. You know what I'm saying? When you talk about, basketball somebody from the LeBron era, we wanna make sure our era stand strong. Man, Jordan, Jordan, Jordan 66, 196 pounds. For show. LeBron is 69270 and jumped the same height. Evolution is a motherfucker. You talking about David Robinson, Kevin direct the same size as David Robinson, shooting from half court. It's evolution. Yeah, we on boss talk one on one. Yeah, we gonna talk.


I'm a big linux fanboy. I think you know this about me. I've created a channel with linux in the name I Love linux and I love most things about linux. I love the choice. I love the fact that it doesn't bug me for updates I love that. I don't have to restart it after I install a program I love many things about linux and I hate many things about Windows and macOS Irrationally so most of the time. I mean if we were completely honest with ourselves We can at least say that if we had to we could use Windows and get work done I would hate every minute of it, but I can be rational about this and say that I could do that But I would still prefer linux 10 times out of 10 and I think the vast majority of my audience agrees with me on that point That all being said there are things about linux that just piss me the fuck off and I'm dropping an F bomb there because it's Necessary so it's late at night It's just about midnight and I've decided to do one of these ranty videos Just because I wanted to do it and this video won't go up for a couple days So maybe I won't even choose to release it, but I don't know The thing I want to rant about today is call is file pickers Now I know what you're thinking Matt How much time do you actually dedicate to your to your thought process to file pickers and the sad fact of it is way too much File pickers are the reason why I can't get my entire family to actually use Linux because the file pickers on Linux are terrible and I'm using the plural bit because there are like nine different file pickers on Linux I mean it's not really nine it might be nine I don't know, but there's definitely three or four of them and they're all atrocious. They're multiple levels of bad Now I'm not talking about file managers like no Atlas or Nemo. Those are fine Varying degrees of fine. I should say but they have a purpose and they are Useful and if you don't like them, there are 12 other ones that you can choose from No, what I'm talking about is file pickers and these things are the things that pop up when you upload or download a file From a program. So if you open up Kaden live and want to open up a project You do that through a file picker if you want to download or upload a file to Firefox It through you know to a website and Firefox you do so through a file picker and they all use different ones and They don't do a very good job of what they're supposed to do. They are functional to the very minimal definition of functional if you know what I mean, so I'm gonna rant about this for Probably three or four hours. No, this video is not gonna be that long But it could could be I could go on for a long time about how horrible file pickers are so Let me go ahead and show you what I mean And I'm talking Softly so I don't wake up half the neighborhood. Oh as it be screaming man. All right Also, I've done this video once before and OBS decided to stop recording So that also pissed me off just a little bit extra more. So I've turned on the Setting info in OBS to ask me if I'm sure I want to stop recording because hopefully they'll prevent me from Randomly stopping the damn recording because that just pisses me off. All right. Anyways Many rant over let's move on to the main rant. Shall we? So here we are in lovely Firefox and Firefox is great. Okay The problems I have with Mozilla aside Firefox remains the Browser of choice for me. I couldn't stand brave. It has some things that just really bother me I couldn't stand Vivaldi, whatever the Point of the fact is Firefox is great But it uses a file picker just like all programs do and this is what it looks like and You probably see this thing probably 10 times a day if you use Linux Everybody does, you know, it's this is probably the most popular of the file pickers. I believe it's based on Thunar or something. I'm not actually sure it's some kind of GTK app. Nobody knows what it is. It's not good. It does have The feature that I think is most necessary when an open dialogue file picker comes up and that search and That search if you noticed that the video just skipped it's because Firefox or excuse me OBS stopped recording again, even though I have it set to ask me if I'm sure and you want to know why because the key binding for for find in Firefox control F is also for whatever reason the stop and Start record but key binding for OBS.  I'm sure I have but anyways a few moments later I've changed that hotkey now. It took me a minute but Now that my OBS problems seem to have been changed or fixed hopefully Let me go back to my main rant which is file picker So if I hit control F over here and that shouldn't you know that shouldn't stop OBS now which it doesn't thank goodness and The search box comes up now. It's somewhere in this directory within another directory I have a image of tux and it has tux in the name of the file. So if I search for tux I Get nothing. Okay. I get absolutely nothing now I can go to a terminal here and I can do locate tux and I actually get Every place where tux is actually, you know, located now. It's not the best search ever Because it's in paths, you know, it's not usable. It's not something that I can drag and drop into Firefox It's of no use for me. Now if I were to use something like Ranger and do search in Ranger I could theoretically go through and enable drag and drop in order to do that from Ranger that'd be fine but if you if I just use the stock standard file picker in Firefox how am I supposed to find the picture unless I know exactly where it's at and Luckily, I'm very organized when it comes to my pictures I pretty much know where everything is and I've only become that way because I know that if I don't know Precisely where everything is on my system and I mean everything. I'll never find it again because Or at least I won't be able to find it in a file picker because the search and file pickers is atrocious Or in the case of something like Kaden live, so let me go show you Kaden live Let's just say I wanted to open up a thing here and we just go to Linux videos or whatever it doesn't matter This is the Kaden live file picker and now they look different because one is a GTK app One is a cute cute app and you would expect them to look different But there is no search in this app as far as I can tell it all now You can't just this is a drop-down. So that's not a search the file name here That doesn't act as a search and if I hit like control f or something That doesn't work. So as far as I can tell there is no search within this at all I thought maybe like if I You know somehow managed to you know get out of the Deactivate the text thing here and like hit the slash key in order to like of them search key binding Maybe that would work that doesn't work. There's no search in it in this at all So if you again, if you don't know the path the exact path and the exact location of the file you're looking for You're screwed you have to go to a terminal or a different file manager in order to actually find the damn thing and That's just done. Now. That's just two of them. Let me show you something else Let me quit this out of here and move this over here. So this right here is Telegram Everybody uses telegram probably or something like telegram and let's just say I wanted to download I did doesn't I think I can actually just download this I can just do this and hit save image s and Oh, actually, this is gonna be different. They've changed it since the last time I was here This is actually the same file picker as Firefox has so that's different than what we used to be for what I'm trying to think of what the other application I was using Was it OBS or not OBS was it Was it discord? It could be it might be discord. Let me upload a file. Nope. It wasn't discord Anyways, I guess it doesn't really matter. There are other file pickers out there There's one that looks really really old that looks like it was from like a Windows 98 style thing And it's not Qt based. It's I don't think at least not a recent version of Qt And there are a couple other ones out there as well They're all terrible that especially when it comes to search. They're all terrible either they're like the cute one and don't have search at all or they're like the GTK one and The search doesn't work. It just it doesn't work. You have to be in the direct if I went back to the Firefox thing and Was in the exact directory that I was but I know the knew the file to be in I could search for tux And it would show up So I don't know what it's using to actually search, but it's not using any of the There's no System-level service that is actually searching and providing that information to that file picker So it doesn't use locate it. Maybe it uses fine or something. We all know find is terrible I mean fine has its uses, but it's not as good as locate For actually searching the whole system. I mean so whatever it uses It's not it doesn't have any kind of index that goes outside of the Directory that it's actually in it just searches that directory and it doesn't search within sub directories It just it's what's not, you know recursive or anything It just searches in that directory for specific files if it's not in that directory you don't find it it's the dumbest thing in the world and We have all these file pickers right because Linux Developers can't agree on shit.  I mean you can just see this I mean this is bad right this is not a good user experience when you want to When you when you have an open dialogue file picker One of the things that you have to be able to do is search and it has to work well It works well on Windows. It works well on Mac OS. It does not work well on Linux and It's not gonna ever This is one of those things that is just going to annoy the crap out of me forever and ever because it's never ever going to be fixed And that makes me sad because like I said at the beginning I love Linux But this is one of those things in Linux that just drives me absolutely bonkers And it's also possible that maybe I just don't know what I'm doing. It's possible I'm an eternal Linux noob and there's a possibility that the cute version has searched And I just don't know what it is. It's possible that there's a key binding that allows you to access search in a cute based Open dialogue file, but file picker or whatever and I just don't know the secret handshake. It's possible It's also possible that there's a command you have to run or something that in order to Enable the gtk one to actually be able to search through things.  I'll freely agree with you. But anyways It's late at night I've had problems with obs enough for the day. I'm done for now So if you want to follow me, you can do so add the Linux cast On twitter and facebook. You saw the facebook page there a minute ago I don't post there very often so whatever you can also support me on patreon at patreon.com Slash linux cast before I go I'd like to take a moment to thank my current patrons devon marcus magland Donnie spin east coast web chris mitchell mr. Fox american camp. Thanks everybody for watching and putting up with this rant I'll see you next time


Welcome back to our last video in lecture seven. Let's look at one more example of using integration to help us with these variable force problems in order to calculate the work. We'll do some more examples of this in lecture eight, but this will be the last one for this official lecture part right here. And so this is the proverbial leaky bucket problem that calculus students absolutely love. Imagine we have a bucket that weighs four pounds and a rope of negligible weight. What I mean here is that the weight of the rope here is so small that we're not gonna consider it. I mean, clearly that's sort of a simplifying assumption that isn't real.  These are things that people do in physics types problems all the time. The thing is we just don't wanna worry about the weight of the rope. We've seen in a previous example how one actually would calculate the work of lifting a rope. So we'll defer to that example if you wanted to. Let's just focus on this bucket that contains four pounds of water. And it's being drawn, the bucket's being drawn up a well and the well is 80 feet deep, okay? Now the bucket's full, the bucket itself is four pounds, but the bucket is contained. It's large enough that it contains 40 pounds of water. And how fast are we pulling it up? It's being pulled up at a rate of two feet per second. But the water has a leak in it. It's losing water. And it's leaking at a rate of 2.2 pounds per second. So assuming we pull the bucket up at this constant rate, how much work is done to pull up this bucket? Now, to be aware that this problem is very similar to lifting up the rope. That is, as we pull it, as we lift it, it's getting lighter so its weight's going down and the force is variable. And so how does one try to accommodate for such a thing? Well, the strategy of accumulation here is to first pick an axis. We have to pick some axis of line that we're gonna integrate along. So for this one, we're gonna choose the bottom of the well to be X equals zero and then we're gonna go upward, go up to be our positive X direction. We could have done it so that the origin was the top of the well. We point down. It doesn't really matter. You'll get two different integrals that actually give you the same quantity. The geometric analysis is what's really important here. And as I did in the previous example, I did X equals zero at the top. I wanna do an example of X equals zero at the bottom. So you see there's not much difference here but you could try it out yourself, see how things are different. And so what we're gonna do is we're gonna take our axis and subdivide it into teeny tiny little pieces, right? And so what we're gonna do is we subdivide and then we have to assume that the problem is easy, which in mathematics the easy, what we mean by easy is we mean it's linear, right? We wanna approximate this with linear approximations. So what do we mean by easy in this case here? Well, we're gonna assume that for this time interval as it goes from Xi minus one to Xi, let's assume that the weight is constant. Let's kind of assume that there's no leakage between these two markers, that when it gets to Xi minus one, it squirts. Then it goes to Xi, it squirts again. Then we go to Xi plus one, it squirts again. Let's assume the mass is only lost and we have those little markers there. Is that realistic? No, but it's a simplifying approximation for which integration will correct any error that we introduce into it. So what we have to do as we work with this work problem, of course, is we're going to work, oh, work is the integral of force times distance, right? So we have to figure out what's our distance function, what is our force function go from there? So our force function in this situation, F of X, it's gonna be weight. How heavy is this thing at location X? Well, there's two aspects to it. There's the weight that comes from the bucket, right? And there's the weight that comes from the water. Now, the weight that comes from the bucket is gonna be constant. The bucket always weighs four pounds. That doesn't change as we go through this exercise. But what's the weight of the water? Well, the weight of the water, it starts off at 40 pounds. And then we lose something, right? Sorry, 40 pounds. Then we lose something. I know that's not super specific, but when you're working with a story problem, it's not like your equations, your functions have to be picture perfect from the very beginning. Just kind of write down as your mind is thinking about something.  What are we taking away? Well, it loses water over time. And so we can say something like, okay, a little bit better than something. We're gonna lose 0.2 pounds per second. Well, you should times that by time, t. So if t is measuring in seconds, seconds maybe since the start of this exercise, right? I want you to start lifting the bucket. So we get this 0.2t. Now that's not quite the right, I mean, that works for us, but we wanna find a function with respect to x because we're gonna integrate with respect to distance, not with respect to time. So how do you write time as a function of x? Well, that's where this rate comes into play, right? So the distance with respect to time is our distance x is gonna equal two feet per second. So if you take two feet per second, and we multiply that by time, which is in seconds, that'll give us how much feet we've done. Now solving for t, this tells us that t is gonna equal x over two or one half x if you prefer. So the distance that we travel divided by two gives us the time. And so if we plug that in here, right? We end up with four plus 40 minus 30. 0.2 times one half times x, like so, which of course will give us a 0.1, I guess it's a 0.1. And so our force function, summarizing what we did right here, the 40 plus 40 plus four is 44. So we get 44 minus 0.1x. This is our force function for which we want to integrate. And this will give us the force, the weight of the bucket at any location. So when you're at location zero, this thing will weigh 44 pounds. When we're at location, say one, right? It would be 44 minus 0.1, so 43.9. Then if you do with the very top of this thing, right? It's 80 feet tall. We could take 44 minus 0.1 times eight. 0.1 times eight, you know, we could figure out exactly how much water is in the bucket at the very end. I don't really care about that too much. Just be aware, that's what we've done right here. So once we have the force function, because we're assuming weight is constant from this right here, these two intervals, we want to go from this interval to this interval. And what's that distance? That distance is gonna be delta x. So now in a position where we can set up our integral that we need to compute, the work is equal to the integral force. Force is gonna be 44 minus 0.1 x. That's the weight. Then the distance it has to travel is dx. Now we're left with the bounds, right? What are the bounds of integration right here and right here? Well, we have to go from the, where can x live here? x could be at the bottom of the well, which represents x equals zero. It can go to the top of the well, which would be x equals 80 feet. And so now we just have to compute the anti-derivative from here. The anti-derivative 44 minus 0.1 x would be 44 x minus 0.05 x squared from zero to 80. Plugging in zero makes it disappear, so we'll just plug in the 80. And we end up with 44 times 80. And then minus 0.05 times 80 squared. Let's see, 44 times 80, that's 3,520. And then 80 squared, that's 6,400 times that by 0.05. That should give you 320, I believe. And so then this adds up to be 3,200 foot-pounds of work. And so notice that the actual calculation of the integral is nothing challenging. There's not hard integrals right here. It's setting this thing up, that is the significant thing, right? Why was the integral this baby right here, right? If you can understand the argument going on there, that's really what we want you to get from these types of exercises. I don't really care about work that much, you know? I'm an mathematician, I'm not a physicist, but many of you watching here are, you know, people who want to be physicists or chemists or engineers or other types of scientists or even math students themselves, right? These applications do have reach into our real lives, whether it's professional or even personal lives as well. And it turns out the calculus is everywhere. We can use it to solve so many problems. It just takes this strategy of take your problem, subdivide it into small pieces. It's like if you have like a six-foot sub that you want to eat, right? You can't just eat in one bite.   And you do this for all the bites there are. You subdivide the problem, solve the easy subdivision because you're adding some little assumption that makes it simple. And then when you take the limit as it goes to infinity, the number of subdivisions goes to infinity, the error will be removed from the problem. It's a beautiful strategy. We'll see some more of this in the next coming lecture videos as well. And I guess it's the end of lecture seven. Thanks for watching. If you like what you've seen, please hit the like button. Feel free to subscribe for more updates of these type of math lectures in the future. Post any comments. Post any comments if you have questions or want to know more about this topic and I'll be glad to talk to you about them. Otherwise, I'll see you all next time. Have a great day everyone. Bye.