As Christians faithful join their counterparts across the globe to commemorate Good Friday, which is the crucifixion of Jesus Christ on the cross, Nigerians are called to use the season for sovereign reflection on the virtues of love, selflessness, and prayers for peace and unity of the nation. Reverend Father Philip Daniel the Catholic priest in Kaduna says Christians and Nigerians should take advantage of the period and pray for God's divine intervention amid various security challenges confronting the nation by keeping hope alive for a new birth of Nigeria. Children, youths, and aged parents were visible to witness the way of the cross service with stations of Jesus' suffering and death. These dramas put many into emotions as they could not hold back tears but reflect on the hope of Nigeria where love, peace, and tolerance reign. We hopeful as Nigerians and let us do the needful as Nigerians by doing that I believe that this country will be better than what it is. I call to Nigerians and especially to our government is for them to do the needful. They know what to do. Let them do the needful and for us Nigerians let us abide by the law and let us keep praying for this country and then do our part as Nigerians. Today is important and it is a job for us to reflect and continue to pray. It's important that we all look into our lives and you know even as a Nigerian don't wait for the leaders. What is that that you can do for this country Nigeria to move forward? How are you contributing?  The Friday of today it's unlike other Fridays of the year and the Friday is indeed good because it's actually through the passion and the debt of our Lord Jesus Christ that we gain salvation. I think that's the apex of our Christianity. Simply means for me as a Christian I actually have the hope of salvation and that's what makes me Christian. At the end of my sojourn here on how it's based on what Christ has done for me.


The conversations that I have, are they just neutral conversations or do they hold the energy of transformation for myself and for other people? Today we're going to be talking about the importance of every single conversation that you have and to actually take you through this journey that we're going to go on together. I'm actually going to tell you two separate stories that have really kind of changed my perspective of life and when I have a conversation with somebody realizing that me but also for you as well that one conversation that we have with someone can completely change their life whether we realize or not and that can be for positive but it can also be for negative. So the first story I'm going to tell you is about six months ago I was walking inside of a parking lot in Austin.  My truck was kind of loud and so I was like what's that and he came over and started talking to me it was him his girlfriend and he's like hey man I just want to tell you I follow you on Instagram and listen to your podcast and I love all the stuff that you do. I was like that's awesome thanks so much and he goes I don't know if you remember we actually met before and I was like really where did we meet and he goes we met in Florence Italy a couple years ago and I was like oh that's pretty awesome and he started telling me about where we met. We met at a pizza and gelato making tour where we were making pizza making gelato and as soon as he said and I saw him and I saw his girlfriend I was like oh I remember you guys we had a conversation while we were making gelato right and he said yeah and in we were talking about you know my girlfriend at the time and I which is now my fiance we were traveling for a month throughout Italy and him and his girlfriend were traveling for like 10 days and he had a job she had a job and he said uh you know he was like yeah man it'd be so nice just to travel and do what we want to like you guys but we have a job and we can't really do that. Well what happened was he asked me what I do and I told him I was a coach and I told him how it worked and I told him how I did at that time one-on-one coaching and how I made money and it was like a probably a 15-20 minute conversation at most like it wasn't that long of a conversation. We had enough of a conversation that when he went home that day he started researching about becoming a coach online and what that would look like and he used to deal with a lot of anxiety when he was younger and went through a lot of self-development to then overcome his anxiety. Crazy enough we had that conversation two years ago never saw him again until I saw him inside of this parking lot and he told me that he after leaving and leaving Italy and coming back to America the day he came back he put his two weeks in at work and he said I'm going to figure out how to start a coaching business because I feel like I have a lot of value to give people and when I saw him inside of the parking lot he said that he now has 11 employees in his company. So I just had a very normal conversation talking about what I do for my profession and how we travel and how we love traveling and we usually travel for you know 30 40 50 percent of the year and he was like I want that I'm going to figure out how to do that and literally he went from that conversation to coming back quitting his job starting his business figuring it out figuring out marketing figuring out all that stuff and now it's not only just his life that has changed but it's also his fiance works it's his fiance now not just his girlfriend his fiance also works in his company and he now employs 11 people so 12 people's lives have changed from that conversation I'm not saying it because I'm not trying to toot my own horn because that's the last thing I'm trying to do I'm trying to say it because I was driving home after that and I was like holy s*** I can't believe how many conversations I have throughout the day and if they have the potential to do something like what happened with this guy and so it was really eye-opening for me another story that I want to tell you that's very similar in a way also happens to be in Florence Italy and it was the very first time that I was recognized by somebody in public and this was back four years ago now five years ago and there was a lady that came up to me and we were walking through a leather shop and she came up to me as we were walking out and it was my fiance and I and she goes excuse me excuse me excuse me and she was like do you uh she had kind of broken English she goes do you uh do you know Rob dial and I was like uh I do I am Rob dial and she's like oh swear to god she goes oh mama Mia and I was like oh my god it's like a movie they actually say this over here and uh and she started saying she's like oh my god oh mama Mia this is I and she started getting like kind of flustered she kept saying mama Mia said like 20 times and she's like telling me how I was running a webinar one time she listened to the podcast I was running a webinar one time and the webinar was like at three o'clock in the morning for her and she woke up in the morning three o'clock in the morning just to be on this hour-long live that I was running and inside of the live I guess I had said something along the lines of like how we need to focus on what we're grateful for and if we focus on what we're grateful for it will change our perspective for our life and she was in and she was on her way walking to she told me a story of how she was walking to um her work that she worked at she was walking across the bridge and she was having this terrible day she got in a fight with her boyfriend and a whole bunch of other things were just happening and she was just having a crap day which happens sometimes right she was having a crappy day and she was walking across the bridge and she goes you know what I'm going to try to think the way that Rob told me to think in that live that I was on and so she stopped when she was walking across this bridge this in Florence called Ponte Vecchio she was walking across it and she looked and she's like oh my gosh what can I be grateful for right now and she's like this this city I live in such a beautiful city like people travel from all over the world to literally spend a couple days in the city that I get to live in and have grown up in my entire life and she's looking over the bridge and there's people that are on canoes and they're going on the canoes and she's like this is beautiful like the mountains are in the distance there's the the river that I'm above right now there's people walking around there's a kayak that's down there as well and it's so amazing that I get to live in a city like this and then she was like what else can I be grateful for and she started listing all of these things that she could be grateful for and she realized that the first half of her walk she was in a terrible mindset and then she shifted her mindset to which she should be grateful for and the rest of her walk she said was amazing and she had an amazing day in it clicked in her head of I am in control of how I feel throughout the day and I say those stories not to once again Tutmar and Hornay sort of way I say this because I want you to understand the power of every single conversation that you have and you start to think about that think about all of the conversations that you have in your life all of them that you've had so far today all of them that you've had yesterday the day before that your entire life and then ask yourself a question the conversations that I have are they just neutral conversations or do they hold the energy of transformation for myself and for other people do I come into a conversation with some form of leadership of hey how can I make this person's day better how can I help them in some sort of way if there's something that I can do to transform this person's day their mindset this moment is there a way that I can show up differently to help this person transform or are we having conversations with people and bringing them down that's something we need to think about right I'll tell you another story what was a huge life-changing conversation for me when I was younger one of my very first mentors I used to run his office when I worked for a marketing company and he's like hey man like we were the number one office out of 740 offices right we were number one in the entire United States and he goes hey man let's it calls me up he says hey let's go get lunch meet me at Chipotle we meet at Chipotle we sit down and he goes all right and he was very brash like just set it straight to the point exactly how was he goes I don't know how else to say this to you um people don't like you and I was like oh okay I wasn't expecting to have this conversation and I was very taken back by it so I was like what do you what do you mean people don't like you he's like listen man I know your heart and I know who you are as a person and I see that come out because I spent so much time with you and I've seen you for so long but a lot of people don't see that a lot of people see this character that you have built yourself into that is not your true self there's a character of this person that you play that is I don't give a damn and you're making fun of and you're you're cracking jokes and all this stuff but what the problem is you're throwing little tiny and what he called them was neggs n e g s short for negative right throwing little neggs at people where it's like little teeny tiny negative comments you know short short for negative comments you're throwing these at them and it might not be one that hurts it might not be two but if they have a conversation and another conversation another one another one another one and even if you're saying it just kidding around they eventually add up and then people start to go yeah I don't really like this guy because every single time that I'm around him he makes me feel worse about myself and that was a life-changing conversation for me to have that conversation because then I started actually thinking back over all of the stuff that I said and realizing that even though I was kidding really what I was doing was I was trying to overcome from a lack of self-worth I was trying to get my self-worth from in little tiny ways bringing people down to make myself feel better than and so I want to I want you to think about this for a second and ask yourself like in the conversations that you have do they hold the energy of transformation or do they have little tiny neggs negative comments in them to bring somebody down just a tiny bit even if it's just a joke like when I used to think oh we're just kidding we're just messing around we're having fun we're kidding we're messing around we're having fun and I started to realize hey this is people are not viewing it this way people actually think that I'm cutting them down and do I want to be known as the person forever that cuts people down no that's not what I want to be and so I started to notice the way I was talking to people and I didn't get out of the habit very quickly but it was something I started working on I would say stuff and be like oh I shouldn't have said that and I still do that now oh god I don't know if that was taken correctly I didn't mean it to a way that it now sounds in my head as I'm starting to think about it from their perspective the thing that I think that we should all think about and the reason why I'm bringing this up as far as how every conversation can change someone else's life is that we should be very aware of the energy that we bring into every conversation because we should try our best to lift people up because there's one thing that I know is that the world is hard enough as it is right now the world's not easy there's some fun things there's some beautiful things there's some really challenging things and there's a lot of people who put on a really good face as if everything's okay and as every as if everything's amazing but in reality they're really struggling and the last thing that I want to do is to accidentally say something to somebody that is not doing their greatest that is not doing the best and I bring them even lower then even if someone's doing great last thing I want to do is bring somebody down I want to build people up in everything that I do and so I want to think to about how powerful every single conversation has the opportunity to be when you go into it with the intention of I'm going to hold the energy of transformation and this is for people who you know even if you have kids think about the conversations that you have with your children the way that you talk to them and the things that you say and the jokes that you might make to your children but maybe they're young and they don't understand that it's a joke and they take it seriously and they take it to heart I remember one time when I was a kid I'll kind of give you an idea I'm in the process of of getting my pilot's license because I've always wanted to be a pilot since I was a kid and I remember when I was younger I was a tall kid and this is probably I was probably 13 12 13 years old probably about 12 years old and I was a tall kid growing up and I remember when I went to the doctor the doctor was like yeah it looks like you're probably going to be over six feet you're probably going to be about six feet six foot one six foot two and I remember I wanted to be a fighter pilot I thought it was like the coolest thing I want to be an astronaut I want to be a fighter pilot and I was really big on like becoming a fighter pilot and flying jets and I don't even remember who the hell this was but I remember having a conversation with an adult one time and the adult said to me well you know that if you're over six feet you can't fly planes right for the for the the army or the navy or the air force you can't you can't fly planes for them because you don't fit inside of the cockpit and I remember going oh man like well if that's the case what's the point of me learning this what's the point of me dedicating my life to learning this if I'm going to a venture I can't change how tall I am and the doctor says because we all think as a kid that doctors know everything it's a lot of adults think that doctors know everything but that's not always the case especially they don't know how tall I'm going to be and he actually ended up being pretty close to right because I am over six feet but the thing that happened though is I've talked with so many pilots now that are just as tall as me or taller than me that fly jets that fly planes and what happened was that one conversation with a person that I don't even remember who they are but I remember that they were an adult and they were older than me so if I have their adult and they're older they probably know better than I do and I don't hold a grudge to that person any sort of way because I love my life as it is and it happened exactly as it's supposed to but how different could my life be right now if that person never said that to me how many people's dreams have been crushed by an adult that says something just kidding ingest that's just side you know side conversation they say something to a child and a child that might not be as intellectually smart as an adult and know that it's a joke or might completely miss it then take that to heart and go oh yeah maybe I shouldn't do that thing and so for those of you guys that have children or you ever around children also be very aware of the way that you speak to them and do you have the energy of transformation or building that child up because you guys remember back to when you were kids like middle school high school those things can be real real hard for people right so the worst thing we could do is bring somebody down that's already down because you know children can be vicious that's for sure we should always have the energy of transformation for everyone that we're in a conversation with and think about how we can build people up as much as we possibly can because you never have any idea how much something that you say to somebody can change your life for the positive or for the negative and so the one thing I want you to be aware of is the stuff that you say to other people because it's important knowing how the world is knowing how people think knowing how tough it is especially right now with all of the stuff that people are going through in the positive face that people can put on how important it is for us to just think about us showing up for other people as much we possibly can and we might be like well yeah but what about ourselves when you show when you show up with the energy of transformation what's beautiful about it and transformation or positivity or building people up is people will usually meet you at the energy that you're at so you can either meet them at the energy that they're at which could be low it could be indifferent it could be just whatever or you can say you know what I'm going to bring people up in every single situation that I'm in every single conversation that I'm in I'm going to hold myself to the highest standard to talking to people the way that I want to be talked to to treating the people the way that I want to be treated and I'm going to have every single conversation that I have hold the level of transformation so that I can hopefully impact as many people as I want to because had I never had that conversation with that guy that was inside the garage I would have never known that that happened had I never walked into that leather shop I would have never known that that happened as well and it makes me think about how many other conversations have I had that have steered people's lives in some sort of way because all of our conversations can steer people in other directions just little teeny tiny conversations and so this really woke me up to the the opportunity that we all have to help people or the opportunity we have where we actually bring people down if we're not being very diligent to make sure we're holding that energy but can you go into every single conversation with the energy of I'm going to go in with the energy of transformation positivity and trying to help this person as much as I possibly can I will not dip into negativity I will not dip into a place that I don't want to go to I want to hold that energy and see if that person can meet me there because every conversation that you has every conversation that you have has the opportunity for transformation hey thanks so much for watching this video if you want to learn even more about master your mind click right here and watch this video as well I think the purpose of this life purpose of us being here that's all I'm here to do there is nothing else for me to do except


Hello, I'm Cyril Petion-Rossillon, or for people who speak French, I'm Cyril Petion-Rossillon. Bienvenue à notre émission. And I'm Lynn Palm, Petion-Rossillon, and we're here to show you some really fun things that will be educational and entertaining, but fun things that will be a little bit different twist to you learning something, because we're going to let our horse be the teacher. And are you sure you want to be a teacher for us? Are you sure? Okay, very good. Well, what we're going to do is go through some fundamentals about riding that we all practice when we ride, whether you're a recreational rider or a show-ring rider. But we're going to show you common struggles that you and I may have all the time with our riding, whether it's with balance or communication with the horse, or whether it's putting the horse in balance of what we want him to do. And we're going to show those common challenges we have and show what a reflection it has on the horse. Then we're going to ride well, and we're going to show the reflections that has on the horse. So we're going to let the horse be the teacher. His name is Lark's Home Run. We call him Homer. Are you ready to go Homer? Are you ready? Are you ready? Okay, very good. And Cyril, are you ready to go? Absolutely, I'm ready. Are you going to be a good student? I'm always a good student. Are you going to be a good bad rider? I'm going to be the best bad rider you'll ever see. Well, let's get started. I see a horse that's got his ears forward, working back and forth, attentive to his rider, relaxed in his tail, a nice swing to the bottom of the tail as he's working. The overall body composure is relaxed. What I really like that I'm seeing is the evenness of the tempo of the trot. Both horse and rider are harmonizing together. Look how relaxed the horse's mouth is. Look how relaxed his overall body composure is. Ears forward, relaxed in his work, evenness astride, and another good tail will always see the bottom of his tail swing back and forth. And a nice harmony to his stride. The proper rider's position and balance are the first fundamental we teach at Pampana Ship Training. It's a little bit like when you learn how to drive. You first learn how to sit, you first learn how to position your hands on the steering wheel, where to position your feet on the pedals. Well, that's the same thing with riding. You need to learn first how to position yourself, how to sit, how to balance yourself before doing anything else. And let's show one of the many common challenges we all have. Let's show the upper body weight. Okay, good. Being too far forward now. Certainly not keeping a good position. When this happens, the lower leg goes back. Note that the horse speeded up right away. But really look at his frowning ears. You can tell that he's not happy there. Good. Now, Cyril, let's go back to a correct position now. Make sure you look ahead. There you go. Look at the ears again. Immediately forward. Look at the relaxation through the neck, the mouth. Again, ears are forward, relaxed horse. Okay, now let's lean too far back and your legs forward.  And again, note the ears, the head, the mouth. The horse is very confused. And as you see, his behaviors are just going to keep getting more and more aggravated. Again, that rider being off balance is always, always changing the demeanor of the horse and his overall expression and willingness. Now, normal position, correct position. Don't hold with those legs. Your legs stay relaxed on the horse's sides in contact with the sides, but legs just having a contact, no gripping at whatsoever. Just a quick tip. Whatever the rider does on horseback will affect the horse's attitude, behavior, but it will also impact his balance and way of movement. And he really proved, I hope, to everyone how much horses love lightness. They love you to give them a signal that gives them a chance to think about it in their head, respond to you, and if you get smoothness, you get willingness. Hey, you've just given your horse the partnership that we're all trying to achieve with our horses. In our next subject, the third installment of our series, Let Your Horse Be Your Teacher, we will show you how the horse's body position affects the horse's attitude, behavior, but also way of movement. Thank you.


Hi, I'm Zor. Welcome to InDesert Education. We continue solving problems which I qualify as non-standard problems, not related to just to check your theoretical knowledge, but just to force you to think about certain things. And the purpose is actually to give you some material to practice this analytical approach to different problems. Math presents a perfect tool to develop your analytical and creative abilities. And obviously you will be using these everywhere in all kinds of industries wherever you work. So mathematics is just a gym for your brain, so to speak. And these problems are the perfect tool to exercise your creative abilities. Okay, so today we will consider a couple of problems. Well, actually it's a one general problem with two different variations and three different ways to solve this problem. Okay, so it's trigonometry. And what is very important is to have certain theoretical knowledge before you approach these problems. So that was the purpose of the course Math 14s presented on the same site Unisor.com where this course is now. This course is Math Plus and Problems and the prerequisite is Math 14s. Over there I do have certain theoretical material about trigonometry and we will also use complex numbers here and others formula. So these are things which you have to basically know from your regular theoretical course of mathematics. And again if you don't anything about these, these topics go back to whatever the source of information including Math 14s course on Unisor.com. So this is for those people who understand perfectly well the trigonometry, the basic course of trigonometry and basic course of complex numbers whatever is supposed to be touched here. Okay, so trigonometry and here is two problems which are actually variations of the same problem.  This is the sum which I would like to evaluate or present it in more compact form. It's sine of x plus k times y where k is from 0 to n. So I have to evaluate that for now. In the regular course of school mathematics you probably learned how to summarize the arithmetic series when there is a constant difference between consecutive members or geometric series when there is a constant factor from one to the next one. This is neither of those. So what do we do? Now before you proceed I do suggest you to think about this, obviously. Now I will present two different approaches to this problem. The second problem actually is the sum of cosines which will be analogous. But in any case I will present two different approaches, one trigonometry related and another related to complex numbers and others formula. So right now I'm going to present the first solution related to pure trigonometry and it's kind of a trick. So whenever you need some kind of a trick to solve the problem question is how you come up with this trick. Well maybe some smart guy did but and since then it's a known trick so to speak. But if you don't know this you have to really come up yourself with some very artificial kind of a trick. I use the word trick to basically simplify this problem to solve it. Now obviously if this is the first problem of this type that's very difficult. I mean you might but anyway again I can actually suggest you a hint. You have to multiply this by something which will drastically reduce the whole thing and basically all members will will cancel each other except two of them which are the very first one and the very last one. Okay that's a hint so to speak right. So with this hint again you can stop the video and try to come up with this particular approach. Again if you don't I will continue talking about this but my problem is that the more problems of this type you will solve the greater will be your repertoire of tricks so to speak. So the next one would be maybe easier or faster for you to come up with certain artificial steps which you have to which you have to make to solve this particular problem. So in this particular case an artificial kind of a step which you have to make artificial in terms of it's not really obvious it's not like according to the theory you have to do this there is no theory but there is this particular trick. So what I'm going to do is I'm going to multiply my sum by sine of y over half. Why? Well again as I said this is a trick which helps and you will see that this helps. How did I come up with this? I didn't quite frankly. Long long time ago when I was probably in high school I read about this or something like this and since that it just came into my repertoire of tricks as I said. So now I basically know that it's supposed to be something like this. I remember that I have to multiply by something from my school years. Now when I was preparing this lecture I was just thinking about what should I really do. I was thinking maybe I should multiply by sine or by cosine by y or by y over half etc. So I just figured around and found that this particular thing does work. So what happens if you will multiply it by this? Well in the short description that would be sum of k from 0 to n, sum of x plus k y times sine of y over half. Okay fine. Now what is this? Okay now remember cosine of alpha plus beta is equal to cosine alpha times cosine beta minus sine alpha times sine beta. If I will put minus here I will put plus here, right? How can I get sine times sine from this? So I would like to cosine to cancel out. I will do the following. Sine alpha times sine beta is equal to, okay I have to take difference that would be plus here and I will subtract sum. It will be minus but with a minus it will be plus and cosine will cancel each other and that would be double so I have to put one half. So this is the formula which is derivation of a simple formula which you have to know from trigonometry cosine of sum or difference of two angles. So I will use this formula like this. So sine times sine this is alpha this is beta so I have to equate it to one half well actually yeah sum k from 0 to n this is one half and here I will put cosine of the difference so it's a cosine of x plus ky minus one half of y minus cosine of x plus ky plus y is it better than it was before? Yes and here is y. This is my sine this is s sine times sine of half of y, right? That's what it is. Now why is it better? Well think about it this way. Let me just get rid of this too. I'll put it here, okay? So let me just consider this sum. The first member is with k is equal to 0 would be cosine of x minus y over half minus if k is equal to 0 cosine of x plus y half. Next with k is equal to 1 what will be x plus y minus half of y, right? So it's cosine of x plus y over half minus if k is equal to 1 would be 3 seconds so cosine of x plus 3 y over 2 plus next one k is equal to 2. 2 minus half is 3 seconds okay so it will be cosine of x plus 3 y over 2 minus 2 and half so it's 5 half. I think by now you realize what happens minus plus minus plus minus plus etc. So what will be couple of last members? Couple of last members will be when k is equal to minus 1 it will be cosine of x plus n minus 1 minus 1 so it will be n minus 3 halves of y minus cosine of x plus n minus 1 plus half that will be n minus half n minus 1 half of y plus and the last one when the k is equal to n I will have cosine x plus n minus 1 half y minus cosine of x plus n plus 1 half of y so that happens this would be the previous one and what remains is this one and this one minus cosine of x plus n plus 1 half and this is equal to this from which our sum of sines is equal to we have to have this one but I can put it 2n plus 1 over 2 y doesn't really matter divided by 2 sine of y over half okay so this is a compact expression of this trigonometric series I call it trigonometric series because it's sines and sines and sines okay so a trick multiplied by this particular factor again you cannot just easily come up nobody if nobody tells you about this you have to come up with this yourself and it's difficult if it's the first time of this particular problem actually there are many problems of this kind when you have to do this kind of a trick to come up with a very easy solution well again the more tricks you learned you read about or you heard about from something the easier will be for you to come up with something new because in in practical life you will always face certain problems which nobody taught you how to solve which means you have to solve them yourself and come up with certain creative solution this is a creative solution so try to do as many of these as possible and mathematics has millions of these particular kind of problems and the more you do the better you're prepared for real life now the second problem which is very similar to this one is with the cosines so I will just use the same approach just slightly different formulas will be so these will be all cosines cosine cosine cosine and this will be also cosine cosine so what do we do to calculate this particular thing well artificial approach is exactly the same you multiply by sine of y over half but now we have slightly different formulas now the first time it was sine by sine which we converted into a difference so every member was after multiplication sine of some sum times sine of this and we converted into difference now we have cosine and sine so we have to you know develop slightly different approach we have to convert cosine times sine into a difference of something okay so here is basically the formula so you have sine of alpha plus beta is equal to sine of alpha times cosine beta plus cosine alpha the sine of beta so this is cosine and sine so we have to extract it somehow from from this well if I will put minus here it will be minus here right so if I will subtract from sine of alpha plus beta if I will subtract sine of alpha minus beta this will cancel each other and this will be plus minus minus the way how I want it actually so I need to subtract it will be two of those so I have to have half of this and that would be cosine alpha times sine beta exactly what we need so this would be so I multiply it by s cos times sine of y over two would be equal to sigma k from zero to n cosine x plus k times y times sine of y over two so this would be alpha this would be beta so it's equal to sigma k is from zero to n cosine times sine cosine times sine so it's a sine okay it's one half sine of their sum x plus k plus one half y minus sine of their difference x plus k minus one half y and now what do we have same thing all the members except except two will cancel each other okay with k is okay this is equal to s cosine times sine of y half all right so this is again I will get rid of this too I'll put it here all right the first member would be with k is equal to zero it will be sine of x plus one half of y minus sine of x minus y over two plus okay with k is equal to one I will have one and a half three quarter three seconds x plus three seconds of y minus sine of x plus one minus one half so it's this y we have minus sine of x is equal to two two and one half so it's x plus five y over half minus sine of x plus one and a half so what happens here it's slightly different but but anyway it's still all out this goes with this this goes with this this goes with this and the one which is remaining is this one with k is equal to n so the whole thing is equal to sine of x plus n plus one half y minus the very first one sine of x minus y over so this particular sum is equal to this divided by two sine of y half and that's the answer now I have suggested two different ways I said I would suggest two different ways to solve this problem so this first way is kind of a tricky you have to somehow come up with this thought to basically do whatever I just said to multiply by sine of y half here is another approach which arguably is more elegant and gives you again relatively fast the result but it's completely different okay now we go to complex numbers and Euler's formula which can which which connects trigonometric functions and exponential function with complex argument i is obviously imaginary one so called square root of minus one if you wish minus one so so imaginary number complex numbers again this is a theory which you have to know including this thing on the course mass routines I do spend some time basically introducing this formula in justification for this formula and Euler is by the way a very famous mathematician Swiss but he spent a lot of time in Russia it's one of the basically founders of Russian mathematics okay so the Euler's formula what does it give you well here it is let's consider let's consider e to the power i x plus k y and have sigma of this where k from 0 to n now according to this that's basically sigma of um cosine x plus k y plus i sine x plus k y from 0 to n now this is the real part this is imaginary part real part comes with real part with all different case and imaginary so I can actually write it as i as sigma of x x plus k y plus i sigma sine of x plus k y from 0 to n from 0 to n so i basically goes outside of sigma and I have basically this is the sum which I was considering the second time and this is the sum which I consider the first time now being as it may I will use this expression instead of this expression because this expression is equal to if I will consider just this one well it's sigma a to the power i x i x times e to the power k y right right because whenever you are multiplying with the same base your exponents are supposed to be i k y exponents are supposed to be added together now e to the power of i x is not dependent on k so I can put it here e to the power i x sigma a to the power i k y k from 0 to n now instead of putting this I will put it this way to the power of k because as you know whenever you are when you are raising something to the power the exponents are multiplying so basically it's the same thing and what is this this is a geometric series with a factor e to the power of i y i i i y and the factor the first member and the factor are e to the power i y so you can basically use the formula for geometric sequence to get the result of this so what you will have as a result you will have e to the power i x times this would be e to the power i n plus 1 minus 1 divided by e to the power of i y y minus 1 so I just used the formula for geometric series it's the last member next after last member minus 1 divided by factor minus 1 in this particular case so this is the answer now it's not looking actually like we had before right why because it's a complex number it should really separate the imaginary part from the real part the real part will be answer to this and the imaginary part will be the answer to this one now how to do this this is unfortunately a little bit more tricky process not tricky no it's more calculation involved which I definitely don't want to do it right now I did it in the notes for this lecture every lecture in on unisor.com has notes so in the notes I really converted into real and imaginary part and I will have and I had exactly the same answers as before here I'm not going to do it I suggest you to do it but the most important part is probably to get rid of the i in the denominator and for this I just multiply it by by e to the power of minus i y minus 1 e to the power minus i y minus 1 both numerator and denominator and if you will multiply this by this you will have only real only real number primarily 2 minus 2 cosine of y that will be in the denominator and the numerator you will just have to multiply and convert it into cosine and sine using the Euler's formula and it will eventually give you the answer so the more elegant this just because it's very fast gives you both sums it's more calculation incentive because it just at the very end when you already have a formula in a complex representation you have to go back to real and imaginary part separately and that will take some calculations again all calculations I suggest you to try to do yourself and again check it against my notes which which which gives you the same answer as I had before so that's that's it for today I suggest you to read the notes they are very detailed and I might actually make a mistake but probably I didn't but anyway the notes are correct so every lecture has both video and the notes on myunicero.com website that's it for today thank you very much and good luck


On to the girls side of things. Faustin squaring off with Kelly or Northam at the Homeark Homes Showcase. First half Greyhounds up one off the inbounds. Lily Manneke hits the 10 footer with a hand in her face. Faustin goes up three. Later in the half, Jacqueline Hubbard cleans up the Emily Curfman miss, finishing inside to put the Greyhounds up five. Faustin would take a 22-16 lead at the half. They'd outscore Kelly or Northam by a lot in the second half winning 55 to 29 in that one. If you've enjoyed this segment of Lakeland News, please consider making a tax deductible contribution to Lakeland Public Television.


All right, sorry for the wait everybody. So welcome to my talk and I guess welcome to supply chain security conference. My name is Adolfo Garcia. I am a software engineer with Jengard, a company trying to solve the supply chain security problem. And I am based out of Mexico City. I am also one of the tech leads with Kubernetes Sieg Release. And I work where I work on the release engineering team leading mostly the new improvements we've been doing to the Kubernetes release process and adding more supply chain security features. I also like to write back around the world and I'm very happy to see everybody here today. I usually don't do these kinds of slides where you present the program to everybody, but today I have like a really packed session because I wanted to give you a lot of information because I want to really understand what we're trying to solve, how we're trying to do it. And I hope that the tools that we're building make sense to you and you decide that they may be useful for you. If I have time, I have a short and a longer demo and I'll try to also cram in some questions, but it's gonna be tight. So let's try to rush through everything. So first I wanna talk a little bit about the challenges of releasing Kubernetes. For anybody that's not aware, Kubernetes is the largest go program in the open source world. It has a vibrant code-based and community-rounded around a thousand people actively pushing comments to the project. We, in secret release, are the guardians of the releases of Kubernetes. So we build all the guidance and the tooling that makes the releases possible every month. And we also manage not Kubernetes itself, but a wide range of other projects that live under the Kubernetes organization. So we have two kinds of releases. One monthly release that involves the batch releases of those branches that are under maintenance, which typically involve high critical security patches and bug fixes. And we also have the big release that happens every three months, four months now, because we switch from four to three, which is the very big release that we do every four months. This release is powered by a team of around 30, 35 people that sign up every cycle to be part of the release team. And they handle everything from bug triage to communications, writing the release notes, making sure the enhancements are well tracked of what goes and doesn't go into the release. And if you want to get involved with the project, it's one of the best areas that the project offers. Anybody can join, no matter their technical level. So as I said, we also manage the tooling that releases the artifacts for over 100 projects that go under the Kubernetes umbrella. Everything from storage, container runtimes, and lots of other pieces that go into the Kubernetes deployment. We release, we support three different operating systems in different architectures under those. We put out over 25 images now, and over 50 binaries ranging from everything from Kube CTL, Kube ADM, the single binaries for all of the Kubernetes components. We produce star bar bundles for the hyperscalers to build the distributions from. We produce also system packages, RPMs and devs, and package the release notes, everything through an automated project process that we've custom written for the project. And important also, whenever we got a new release, parts of the Kubernetes code get spun off into repositories, which are some of the libraries that are used in a lot of cloud native projects, like for example, Client Go. Whenever we tag the new release, those projects get tagged as well. So I hope that gives you an overview of how much stuff goes into a Kubernetes release. And this is the artifacts that we put out are consumed downstream about thousands of users. So we take our responsibility very serious. So we started adding some security features to the Kubernetes release process about, I started working on it about three years ago, adding some of the common things that you now find in every supply chain security problem. So I want to talk about a little bit about why supply chain is a big concern. So when you think about other areas of Infosec, you'll see that those areas have come up with solutions that have been tailored to the particular challenges. So for example, if you think about network security, we have come up with solutions such as making our networks a tiny impregnable fortress where nothing goes in and out without supervision. And we wrap traffic in cryptography so that people cannot snoop on what's going, what's traveling through our networks or forge the origin of that traffic. And supply chain security has its own set of particular challenges, which are, well, we'll see what those are. And I want you to hold on with me because I'm gonna try to rush through all of this. So let's start with this. So this is you and your code. And typically the way we work today is that someone, a developer, will push changes to a code base. And those changes need to go out to the world. They need to be released. The new features need to be used by users. So if you're doing anything meaningful, you're most likely pulling dependencies that can come from your language or form a third party that is some public library that you're pulling into your code. Most people will also host their code in a version control system, most likely Git. And in order to build or release your project, you'll most probably have a builder. Builders can take the different shapes and forms from a simple bar script or make file all the way to a sophisticated release process like Kubernetes has. And if you think about it, builders are also code that gets executed arbitrarily whenever you run and whenever you release or build the software. So that introduces a new set of dependencies into your project, which have to be tracked as well. Then typically your builder will be running inside of a build system, something like GitHub Actions or GitLab. And unless you're, of course, you're laying your project away and releasing from your laptop, which you probably shouldn't do. And whenever you run the builder, it'll put out some artifacts. They can be binaries, they can be system packages like RPMs or devs, they can be container images and lots of other things. And finally, adding to the complexity of this, that there's a human element. So most people don't work, especially if they had a setup like this. They don't work by themselves, they are part of a team. Oh yeah, container images, they also pull another set of dependencies, the operating system dependencies that make the container run. And as I was saying, the other part is the team. People are not alone. They typically are part of a team with more people. So of course, this is a super simplified view of the software release cycle or development loop, but it will serve as well to explore some of the challenges in where your project may be open for attack or become vulnerable simply for the things you're pulling into your code base. So the first one is you can, for example, pull in a dependency which maybe have a vulnerability in it. And this vulnerability may come from an attacker injecting malicious code into the dependency or a well-meaning bug that some programmer introduced on inadvertently upstream and you pull it in. This is like your log4j situation, for example. Then the next one, an attacker could go and breach your version control system. If you remember there was a hack, I think in the PHP project where an attacker was able to breach into the Git server of the project and start pushing malicious commits into the code. Luckily, it was caught really quickly and the Git server secured. The next one, you can attack the build system. If you've seen the wonderful demos of that people like Doffy Cooley, Ori and Coldwater, Roy McCoon, all of those folks do about attacking Kubernetes clusters, those techniques can also be used to own your text on deployment or your Git lab runner. And once you have control of the node, you can pretty much make it do whatever you want. You can attack your builder, especially sometimes builders are left in another repository forgotten and not very well taken care of. If you take control of the builder, you can also make it produce anything that you want. For example, binaries with malware in them. You can attack the system package builder. So for example, imagine an RPM build command modified to inject malware into RPMs or same thing with the container image builder. So you can potentially envision a Docker daemon modified to inject layers of fun stuff into every container it turns out. And finally, if everything else fails, you can always attack the artifact distribution network. I think there was a case with Sourceforge where one of the mirrors was compromised and that mirror, I think it was the Korean mirror, started distributing versions of PHP, MyAdmin with malware in them. And finally, there's a rogue element in your team. Everybody thinks that their team is super secure unless until you something that happens without this rental employee or someone that wishes to make a statement with your project. And that person happened to have push access to your code and decides to inject something unwanted into the code base. And when you consider all of this, it's kind of scary. And then if you're a software developer, you have to being aware of all of this, turn back to your client and say, hey, it's okay, you can run this new version in your nuclear reactor or whatever, it's perfectly fine, go. And it's a scary thought, but that's not the worst part of it all because if you consider this system, this thing that we just saw, is simply just a component of our giant network of projects, each feeding into their artifacts and libraries into the next one, into the next one. And if you think about it, it's a giant mesh that simply defies human comprehension. And the scary thought is if you think that the compromises that we just saw could happen in any of these projects, then that vulnerability can flow to any of them. So it's a scary thought to just be aware of what's happening. That's why we take the same supply chain really, really seriously and we're trying to put to work the latest techniques that we've been finding. And even when you see this, people are still going and saying, okay, sure, run this version in your financial institution. It's perfectly fine and people will just go and do it. So we are in a really delicate case right now in a really difficult situation because we still have the supply chain pretty much insecure and we're trying to come up with fixes and making sure that everybody's doing their part to secure it. So as I said, it is scary, but there's hope in the way. And so as I was saying in the beginning, we are trying to come up with techniques on the supply chain security space to really alleviate this a little bit. So what's the first one? Well, the first and most important thing that we're trying to do right now is to gain visibility into what's happening in the supply chain. And so what we do is that we take these built environments that we have come up with. And we start if you think about an artifact and you want to make sure that it's safe to run, you need to make sure that you can answer some questions about it and some questions like for example, if I take an artifact, what's inside my artifact? What is it made of? What are the components? What does it look like inside? Do those components have to be compromised? Did it come from the repository it's actually supposed to be coming from? Who committed the latest changes to the artifact I'm using? What was the version of the builder? Because if it's an unexpected version, it could be a sign of trouble. What's the build system that produced it properly secured? Who was, how was the builder invoked? Did I pass any parameters to it? Or better yet, tell me, you didn't pass any parameters to the builder. And are these the expected artifacts? So do they match that part of the build process where they were produced or were they modified in any way? And answering, if you want to do a good determination, whether it's safe or not to run one of these artifacts, you can answer at least some of these questions. The more you can answer, the better. But the problem is they start piling up. And all of a sudden, you end up with lots of questions that you need to answer. And if you think about rapidly shifting workload as we have, for example, in containerized workload in the cloud, they start to become a data problem. Because potentially every new version has a different set of answers. And it's too much. But yeah. But the good thing is that machines are here to help us. And so we have new tools at our disposal and new manifests that are machine readable, which can give us that information. So the first one is the manifest that answers what is inside a piece of software. And it's, of course, the software wheel of materials or the S-bomb that we so lovingly like to call it. A software wheel of materials is a list of components inside of a piece of software, which will give you lots of information about it. What are the components? How does the structure inside of it look like? What's the licensing? How do the components relate to each other in that software project? And even some information that isn't detectable directly in the artifact itself, like what were, for example, some of the tools used to build it. The second question, how was this software put together? And that's why we have proven and set the stations. This help you answer the rest of the questions. So just like a short break here, when I talk about proven and set the stations, I'm talking about internal salsa proven and set the stations because there's, I don't want to sort out the holy words about what an attestation or what provenance is. So when you have an attestation, it'll give you information. An attestation is composed of four key pieces of information, which is what were the materials used to build my piece of software? What was the transformation done to it to build it? Who did the actual build and using what? And what were the artifacts that came out? So if you complete those four pieces, then you can answer a lot of the questions that we just saw about the artifact flowing through the release process. So once we visited those two, I want to give you an overview of how we are trying to build a very salsa compliance in Kubernetes. So salsa, for those of you that are not aware, is a framework that helps you do gradual steps to harden your release processes. It'll give you information and assurances about the artifacts that you're putting out that they have been built in a secure environment. So this is the acronym supply chain levels for software artifacts. I actually wrote it in big letters because I always screwed up. And then let's see a little bit of what we're building here. So the discussions to build these newer software artifacts in the Kubernetes process started around 2020 at the end of 2020. And the first project we were to was building the Kubernetes system. As we saw at the beginning, Kubernetes huge project. And we put out a lot of artifacts which need to provide their users with some visibility into what went into them. So we started building the S1. Kubernetes not only is a big project in terms of the artifacts it puts out, but it also pulls like I think it's around 240 dependencies, which are all listed in the S1. And so if you use the S1, you can know which appendices, the versions, and the licensing information for all of them. And we started adding an S1 to the releases in June 2020, 2021. And then on October during KubeCon in Los Angeles, we announced that we were doing the last missing bit to take Kubernetes to Salsa 1, which was providing SAS attestations of how we were building the software. At that point, we added the SAS attestations inside of the release build, which is something that you should not do. That's why we built some of the projects that you're going to be seeing today. Then in March 2022, we finished setting up the signing infrastructure for the project, which meant we had to set up a bunch of cloud resources to get identity and hardware and other cloud resources to make signing possible, not only and also modifying our tools to support signing. We partnered with the Sixter project, which for those who are not aware of Sixter, it's the best way to sign the software, your software project out there. Highly recommended that you go and check the project if you want to start signing your artifacts. So when we finished modifying our tools to add the Sixter libraries and making them into our processes, we started signing the container images in last March. And but this also meant that we have in place the necessary libraries to start signing other kinds of artifacts. For example, we are about to start signing the binaries, and we are about to start signing the attestations that we produce. And planned for this October for the next KubeCon, we are trying to do some Salsa level three-ish. And by level three-ish, I mean we're going to do the best that we can to get it to Salsa level three. And this is because Salsa level three imposes some requirements on the build environment and the infrastructure that some of them we do not have control of because it's shared with some of one of the parent company supporting Kubernetes. And so we are going to do the best that we can, but we are still deciding. We're also shifting a lot of things inside the project, so we'll see. But we are going to definitely do sign attestations, for example. So as you can see in the Kubernetes case, the best way to try and start your Salsa journey is to do it gradually. So when we started in provided artifacts to comply with level one, it wasn't perfect. It is not perfect to this day. But at least we were able to start giving our users at least some visibility into what's going on into the release process. Yes, it's falsifiable. It's not the most secure document in the world. But if you choose to at least take a look, it gives you that visibility. And not only that to our users, but it also gave us some visibility because at some point we were at a point that we didn't even know what we were releasing. So we now have full documentation of everything. The second one is the anti-temporal artifacts. So when you have attestations for what it's happening in your release process, and you sign them, that means that as the artifacts flow from one stage of the build to the next one, they cannot be modified because you have an attestation which has all the hatches of those artifacts. And since it's signed, you cannot modify it. Salsa 3 provides assurances that the infrastructure has been hardened. And as I was saying, the project is trying to do its best with the access that we have. And we'll see up to what level we can get it. And level four is that mythical level which is very hard to comply with, which we are not even trying because in that case, since we're open source and some parts of the infrastructure are of control, we are not trying. So in order to do this, we built some tools. And the first one, that came out of the S-Bomb effort. So we built a lot of custom tooling to produce the Kubernetes S-Bomb. And the first one is BOM, which is a tool that lets you create a software build of materials in SPDX. It's now being updated to work with SPDX 2.3. And it has a few features that are specialized, mostly for containerized workloads. And those features have been tested really well until, because it's been battle tested inside of the Kubernetes project. It has already over 120 starts in GitHub, which is, I don't know, at least the biggest one, the highest number I've achieved. It has a base of 25 contributors and supports generating your full dependency for Go projects, which is what most of the cloud uses, the cloud native landscape uses. It's been, it's in use in several well-known projects already to generate their S-Bombs. And it has some cool features like visualizing S-Bombs. As you can see here, it'll give you, you feed it an S-Bomb and it'll show you how the information is structured inside. And it also has a tiny query language where you can feed it an S-Bomb and ask it for things like, give me all the Debian dependencies that you can find there. It'll do that for you. And it's also now in incubation in the automated compliance suiting attack of the Linux Foundation. But I've done lots of talks about BOMB. You can look them up on YouTube or ask me questions if you want. But the one I want to show you is this new one, which is called Tejolote. So for those who are not Mexican or not Mexican enough, the Tejolote is the handpiece of the Mexican mortar used to make salsa. So you take the mortar, which is called the molcajete. Tejolote, you grind your tomatoes and chilies and other delicious stuff in it and you get salsa out of it. And that's why you named it Tejolote. Also because I wanted to have fun with people trying to pronounce that. And the features is, of course, it's new, new, new because it's, right now it's under test on the Kubernetes release process, but it's already working. And right now it's in shift from my personal GitHub to the Kubernetes organization. Maybe by the time if you look it up, it will be under the Kubernetes organization already. So it works, it's API based. So it's an attestation. If you can make up the information, it's really not very trustworthy. So it works by querying your build system to extract the information it needs. Supports GitHub Actions. It supports Google Cloud Build and pretty soon it'll support Proud, which is the CSED system that the Kubernetes project uses. It has sleep and resume capabilities where you can store its state in a file and resume after the build is done. It also supports sending invoking messaging itself in pops of messages. It supports, of course, signing the attestations natively with Sixer and has a bunch of artifact collection back and switch I'm gonna show you in a little bit. So the way it works is like this. You have some source code that fits into a build system where it triggers a builder and it outputs some artifacts. I intentionally left the build system black because we're treating it as a black box. So we will only tell you what the build system told us it happened. We don't make any assurances of the quality of the build, but we'll at least record what went on in there. So the first thing that the whole thing will do is it will observe the source code as it flows into the build system. It will record things like, for example, the build point, the Git commit where your project is building because that one has to be set forth, has to be registered in the attestation because if you have an artifact, you wanna make sure that you can go back and know from which build point it was compiled, for example. So once it observes all of the inputs to the build system, it will watch the build run. So you can leave the whole of the running and it will query the API every few seconds to watch as the build runs or you can put it to sleep with the information it gathered and resume it at the end. Once the build is done, it will query the API of the build system and it will collect the artifacts and it will collect information it needs about the run. So which builder was used, from which repository, the version, the identity of whatever run the build process and so on. And finally, it will collect the artifacts. So to collect the artifacts, it uses a number of backends, which I'm gonna go through. The first one are the native artifacts of the build system. So cloud build and actions, both support storing artifacts natively. It can use those if you want. If you don't specify any, they will, those are the ones that will be used. You can tell it a directory to look for artifacts so everything it finds in there will be recorded in the attestation as artifacts. It can read from buckets currently cloud storage and S3 is almost ready. And it can also query OCI registry. So the way it works is you point it to a registry and before it runs, it will index everything inside of the registry and after the run, it will index it again and the deltas are gonna be reported as artifacts of the build. Finally, well, not finally. It also can use an attestation. Why? Because if you have as a multi-stage build, you can have, you should be attesting to every of those stages. And if you're producing attestations of each of them, you can feed the attestation to the Holote to build, to attest the artifacts that came out of that stage. And finally, and this is the one I'm most proud of, if your build produces an S-bomb like it should be doing, you can feed that S-bomb to the Holote and it will record the artifacts that came out of the build using the S-bomb to use as subjects in the provenance attestation. Finally, it will take all of that information, build the attestation and sign it with Sixer. I may have, since we started a little late, I'm not sure if I'm gonna be able to make it to the demo, but let me try. So, luckily, I have a short demo, which is this one. Yeah, so this is the short one. So this demo, what it does, it's just a Go program. And the only thing that it does is it prints out, I'm ready to be attested. It's just a binary. And the way it's set up, it has two workflows in GitHub Actions. The first one is a simple Go builder which will output a name binary called Hello and it will store it with a name binary in the GitHub artifacts. So let me trigger this one and I'll show you the other workflow. Okay, so it's running. And let me show you the other workflow that actually builds the attestation. Oh, sorry, that's the other one. So this is the attestation. We currently do not have an action that you can simply put it in, pull it, but this works, it basically pulls down the code, compiles it and runs it. The invocation is here at the very end. I don't know if it's big enough, probably not. So it's the simplest invocation of the demo. It's called the binary, the holota test and point it to the build, which is past the workflow run here and I'm telling it to sign it. It will use the OIDC identities from GitHub Actions to do the signing. And if everything worked, we should already have, yeah, it's building the attestation. So it builds already the project and it's doing the attestation and it's done. So this actually was triggered after the build was done. We didn't need the multi-stage because it's such a simple project. And right here, it actually took all the information it needed from the GitHub API. This is on the development, so it's had a lot of output here. And basically at the end here, it collects all of the artifacts here. Yeah, this is it. So it gets information from the GitHub API and detects the artifacts in the native artifact store of GitHub Actions. And well, it's signing the attestation here and you can see it, this is the actual signed attestation. It's basically for encoded, so it's difficult to understand but I have it copied here so that you can see some of the parts of it. This is almost the last slide. So this is, it has the subject and you can see here the binary that I specified in the build. It has the predicate with all the information and the repository where the builder was and well, there's a bunch more information below that, which I'm happy to show everyone after this if you want. And yeah, so this is basically it. This was the simplest configuration of the attestation but you can use it to attest to more complicated builds. And I had a, yeah, this one. So if you want to get involved in developing any of this, please come join us in C-Release. We're always happy to see new friends show up on our calls. You don't have to be an expert. You don't have to know anything. Just show up and say hello. We meet every Tuesday and come by, say hello. I can introduce you to the code base. I'm happy to do workovers and I don't have any more time for questions but I'm happy to answer any after we are out of the session. So thank you.


Hello and welcome to the session. In this session we discuss the following question which says evaluate the limit limit n tends to infinity 1 upon 2 plus 1 plus 3 upon 2 plus unfair on plus n upon 2 this way upon n square upon 4 plus n plus 3. Let's now move on to the solution. We are supposed to find the limit limit n tends to infinity 1 upon 2 plus 1 plus 3 upon 2 plus n sir n plus n upon 2 and this will upon n square upon 4 plus n plus 3. We are given as limit n tends to infinity. Now taking 1 upon 2 common from the numerator we get 1 upon 2 into 1 plus plus 3 plus and so on we hold and this will upon 4 common from the denominator we get 1 upon square into n square plus and this is equal to limit n tends to infinity 1 upon 2 into now 1 plus 2 plus 3 plus and so on plus n minus this sum is equal to n into n plus 1 the whole upon 2 and we can write n into n plus 1 the whole and this will upon 2 and this will upon 1 upon 4 into n square plus 2 the whole. Suppose that we have limit n tends to infinity 1 upon 4 into n plus 1 the whole would be plus n the whole and this will upon 1 upon 4 into n square plus 1 the whole. This 1 upon 4 1 upon 4 times this and so this is equal to limit n tends to infinity 2 upon n square plus 4n dividing action which is n square so the value in the numerator and denominator by n square limit infinity 1 plus 1 upon n this will upon 1 plus 12 upon n square n to 0 upon n square with n to 0 therefore this limit would be equal to 1 plus 0 since 1 upon n would be 0 this upon 1 plus 0 so this is equal to 1 quiet limit which is infinity 1 upon 2 plus 1 is equal to 1 but we have understood the solution of this question.


Don't let the narcissist intimidate you. Don't let them frighten or threaten you into doing what they want you to do Maybe you have left the narcissist Or they have discarded you And you've developed these feelings of worry nervousness or an ease They're not even around you anymore But you still feel as though you have to check in with them you shaped yourself to comply with their rules or standards To be how they want to do to be rather than how you wanted to be Not because it was what you wanted to do But because you just wanted to prevent any trouble or distress You wanted to preserve the settled and peaceful condition You wanted to make them happy or satisfied because no matter what you do the narcissist always makes it seem like it's not enough Like it like there is nothing you can do to satisfy them, but that has nothing to do with you It doesn't mean that you're not enough They just have an insatiable desire that is impossible to satisfy and this is what causes you to prepare or train yourself into developing these patterns of behavior of Trying to make the narcissist feel more comfortable Trying to relieve their stress or tension because you just want to avoid conflict You want to avoid any arguments or disagreements But they're always complaining They're always expressing their dissatisfaction or announce about something they have this pessimistic attitude Where they tend to only see the worst aspect of things or they believe that the worst will happen They always have something negative to say And then it affects you it affects how you feel it affects how you look at the situation and then before you know it You're suddenly becoming negative and dissatisfied with everything This is why you have to distance yourself from people who constantly complain People who constantly express their dissatisfaction were knowing about something People who always have something negative to say When you're constantly around someone who complains about something in a persistent irritating way It will cause continuous trouble or distress to you It will cause you to develop anxiety or depression Which is why you should avoid these kinds of people Just like you would avoid the coronavirus Because if you're around it long enough You will get infected Then negative feelings or ideas will take hold of you and You will become contaminated Until you will then have to quarantine yourself Because you have been exposed to an infectious and contagious disease Something that is likely to spread or influence other people This is why you must avoid these types of people at all costs You have to reverse these patterns of behavior of trying to make the narcissist feel more comfortable Walking on eggshells Be an extremely cautious about your words or actions You have programmed yourself to say or do things in a certain way To avoid making the narcissist angry or upset You have suppressed so much of who you really are To the point where you have now become this quiet gentle character Who is easily imposed on easy to take advantage of By demanding their attention or commitment and you have become very submissive You have allowed yourself to be controlled by the narcissist You are ready to conform to their authority or will You have become obedient or passive Willing to comply with their order or request You are accepting or allowing what happens what the narcissist does to you Without active response or resistance Now you have become a recluse You have become a person who lives a solitary life and tends to avoid other people You prefer not to associate yourself with anyone You have become shy or introverted You're no longer readily revealing your thoughts or feelings You've become reserved or withdrawn You don't want to communicate with other people It makes you feel nervous or fearful When that is not the way you should be conducting yourself That is not how you should be But even after you have left the narcissist It's not so easy to go back to being the person that you were before And the longer that you stay in this state The most susceptible you will be to another person who is like that Because that's exactly the type of person they're looking for Someone who is easy to control Someone who is easy to take advantage of You have already been prepared or trained For the purpose of being that type of person Become aware of how you have been conducting and presenting yourself Remove the fear Stop trying to comply with their rules or standards Stop holding yourself responsible for the narcissist's peace or comfort Because as you're moving forward This could affect every aspect of your life The more time you have spent around them The more difficult it will be You may find yourself still conducting and presenting yourself as though they're still around As though they're still a part of your life And now you've become this quiet, gentle character You have become obedient or passive And you have been molded to be that way Through the recurrence of actions or events Of trying to please or satisfy the narcissist It causes you to doubt yourself Where you no longer trust yourself enough to make your own decisions You will always need someone else's input or opinions Or you will analyse things too much before making a decision Sometimes you won't even make a decision Which only leads to a lack of activity, growth or development Because you are used to the narcissist's forceful arrogant behaviour You are used to them laying down principles as undeniably true Their fixed views and preconceived ideas Their criticisms It caused you to overthink and over analyse the situation It caused you to be very critical of yourself To ensure that you didn't make a fault or mistake Because you didn't want to experience any arguments or disagreements with the narcissist Regarding a decision that you have made You were constantly walking on eggshells Constantly on edge In fight or flight mode And this is what causes CPTSD Complex post-traumatic stress disorder You became used to living in situations Where you were constantly under stress or tension You were unable to relax because of anxiety or nervousness And now it has made you want to avoid any pain, danger or uncertainty You had so many arguments and disagreements with the narcissist That now all you're looking for is peace All you're looking for is freedom from any disturbance Now you're doing whatever you can to avoid any arguments or disagreements Or some of you have become so hypersensitive to disagreements or criticism That now even the slightest problem or issue Is causing you to become easily hurt, worried or offended It causes you to become hyper-vigilant Where you are highly alert to potential danger or threat When you were not like that before But when you have been around the narcissist or toxic person for too long It will produce these types of effects within you You have to take the focus off the narcissist And start putting the focus on yourself Put your time, energy and attention into yourself Love yourself Focus on your growth and development as a person You need to be a priority in your own life You need to treat yourself as being important Take away the self-doubt Start believing in yourself Start trusting your own intuition Whatever you feel passionate about Whatever you have strong feelings or beliefs for That's what you need to do You may not succeed the first time But it doesn't mean that you should give up You just need to make some improvements or adjustments Instead of feeling anxious or troubled about everyone else Do something differently and start worrying about yourself Don't overestimate yourself You can't just go around supplying all of your time and energy To those who are in need When you need yourself more than anyone else When someone is very selfless and always focusing on everyone else That's one of the biggest signs that they are neglecting themselves They are more concerned with the needs and wishes of other people Rather than with their own Be willing to take risks Be confident and courageous Don't let the narcissist control your life If you show them that you're afraid They will continue to keep you in that state Sometimes you have to use their own tactics In order to defeat them Sometimes you need to be the one Giving an unfavorable response You have to show them That you're not going to be easy to take advantage of Don't let them intimidate you They bark But most of them don't bite They're full of big talk But they lack in action, power and substance They will give you cut and remarks To make themselves feel superior or powerful But they actually have a soft personality underneath They make empty threats or promises They put on an act to deceive you to their abilities or intentions Most of the time They're just trying to impress you By affecting greater importance or merit Than what they actually possess Be brave Be strong Show courage Stand up for yourself Defend yourself Be strong when facing a conflict Defend your opinion or point of view Fight for yourself And don't wait for someone else to fight for you Thank you for watching I hope this video resonated with you Please like, comment, share and subscribe Click the bell icon to receive notifications for my future videos If you like to donate My PayPal link is in the video description Coaching inquiries you can even meet At natsoficoachingatjima.com Thank you for watching and I'll talk to you soon


Moving operations have one thing in common. They don't take much time in one spot, usually less than a day. As far as the public is concerned, you're gone before most people realize you've even been there. That's not so bad considering that your exposure to traffic isn't very long, but there are a couple of hitches. The fact that you're there at all is going to surprise the few drivers that do come your way and it only takes one to mess up your life. The other hitch is that your traffic control needs to be portable. That means you're not going to have the protection like you'd have on bigger, longer jobs. But every situation is different, so we're going to look at some typical moving operations going from the least dangerous to the most dangerous. Probably the least dangerous moving operation is the one that has the most accidents associated with it. Luckily, you're going to have the protection of a heavy vehicle moving along with the traffic. Snow removal is different than any other kind of moving operation.  That's why, in addition to your normal checkout procedures, you have to make doubly sure everything's working right. Visibility is your biggest concern. Windows and mirrors need to be kept clear, both on the inside from fog and on the outside from snow. And it's especially important that others can see you. You're not going to be working under static traffic control with sandbag signs warning everyone you're out there.  That's why all your lights need to be working. You need to be visible from a good distance, especially in bad weather. That's why everything should be working before you leave the shop. You don't ever want to get caught unprepared in a blizzard. At night, visibility for you and for other drivers is even more important. You need to keep your eyes on the road. That means you should be able to find and operate all your controls by touch. When you're taking up more than your lane, trying to uncover the centerline, you're not giving other traffic much room to pass.  If you do change course, make sure you don't go too far in the opposite direction and hit something on the other side. Like the night, snow cover also makes things hard to see, like curbs and islands. Anything you can't see is real easy to hit if you don't know where it is. The only way to know where it is is to memorize its location before the snow falls. It's too late to do that now. You're operating under a lot of pressure. You're in a big rig that's hard to control. With a wing, you're wider than a full lane. It takes you longer to stop and it takes a lot of room to turn around. You don't want to back up either because like in most trucks, you can't see very well behind you and you're not going to have a spotter to guide you. Putting that truck in reverse is asking for real trouble. In addition to just doing your job, you're going to be dealing with traffic too. They're going to be speeding to get home and out of the weather. Your driving can be real frustrating to some. Everyone's following you sometimes blindly, sometimes too closely. All the more reason you need to be alert to what's going on around you. All these examples assume you're going to have the right traffic control setup and that it's working as it should. Other than traffic control, one of the most important safety factors is whether or not you're in a vehicle. Vehicles offer you some measure of protection. You're surrounded by a lot of heavy steel. You're big enough to be seen from a distance and you're moving in the direction of traffic, although not as fast. Your large size and movement may give fast moving traffic a few extra seconds to react before they're on top of you. If you get out of your vehicle, however, you lose all these safety features. You lose the protection of your vehicle and the safety of your motion. If there's no traffic control, you lose a lot of your early warning too. Your main protection now is you, your eyes and your ears and whatever bright clothing you might be wearing that might be seen at a distance. Hopefully, you won't be out there very long, but that's up to you too. It's never a good idea to travel opposite to the flow of traffic, but sometimes you have no choice. Here, there's no escape route. Your only protection is your traffic control and the fact that you're in a heavy vehicle. But if an approaching driver makes an error in judgment, you're the one that's going to pay for it. As you can see, how close you are to traffic is very important, but it's not the only safety consideration. You can be in a vehicle and well off the road and still be in a dangerous situation. Steep slopes can turn you over before you know it. Roll bars and seatbelts are musts on any kind of steep terrain. When you're on foot and off the road, your exposure to traffic may be limited and that's good, but for short periods you might be pretty close to traffic if not on the road itself. In these cases, you're at the mercy of the traffic. They're not expecting you to be there, nor are you big enough to be easily seen at a distance. If you're not paying attention to where you are, you're going to be like a sitting duck. If you're on the wrong side of a guard rail, you just might get trapped. Speaking of guard rails, if you're working for a longer time with a larger crew, you're getting into a more dangerous situation where you'll need to close off a lane. If you're replacing a guard rail that's been damaged by vehicles hitting it, think about that for a second. If they hit it once, do you think they'll probably hit it again? Maybe when you're out there? Don't forget, if you're on foot, there's no escape route, except back into traffic. When you move out into the traffic lane itself, you're moving into a much more dangerous area because this is where drivers usually drive and they're expecting to have a clear path ahead of them. This is where flaggers and other traffic control become so important. If you're on foot, working near the center line, you have little physical protection because a temporary barricade or cone isn't going to stop a vehicle from entering your work area. So you've got to keep your mind on where you're at. From here on in, the longer you're on that road surface, the greater your chances of getting hit. Once again, you're the sitting duck. When you add more equipment to the work zone, you're adding even more hazard to an already dangerous situation because now you have to worry about your own traffic coming inside the work zone. Your traffic controls won't work for you in here and trucks and big equipment operators can't see much around them. Plus, it's hot, it's noisy, and it's dusty. All distractions that keep you from thinking about where you are and what you're doing. Patching is a big job. You're going to be out there over a long period with a bigger crew, more operations, and more equipment moving around you all crammed into that same narrow lane. Plus, you have the noise, the dust, and the pressure cooker job of getting it done on time so you can get out of there before nightfall. This is where a pilot car that's physically controlling the speed of traffic makes the situation a lot safer for you. But on an interstate or four lane, the traffic speed is not controlled. You might have a slow power, but it's like asking a charging pole to slow down. A sleeper could enter your work zone and be on top of you before you know it. The two maintenance workers who were killed in this accident probably didn't have a second to get out of the way of this semi when it barreled in. They probably never saw it coming. The only way traffic on the interstate is going to slow down is if there's too much of it and it starts to back up on itself. That reduces speeds in the work zone, but it only pushes the danger of an accident back up the line where traffic is still coming at a good clip. Back in your work area, you're not so safe either. As the traffic count rises, so do tempers. Frustrated and angry drivers can be very unpredictable and dangerous. Ultimately, your best protection is yourself. Your eyes and your ears and your common sense about what could happen in any given situation. Checking out the situation doesn't take long. Every so often just a glance or two over your shoulder to remind yourself of what's going on. Check out how close you are to traffic. What is its speed? What is its visibility and being able to spot you from a distance? What kind of physical protection do you have separating you from equipment and traffic? If everyone just keeps an eye peeled every once in a while, then sometimes someone just might be watching to give you that extra few seconds warning that could save your life. The more you're aware of what's going on around you, the greater your chances are of not being a sitting duck.