Sunday, May 17, 2009

3A MAT recursive formula, AP's & GP's

Those of you attempting Exercise 8A without students with a thorough grounding in AP's & GP's in year 10 will be scratching your head at this chapter.

Q1: creating recursive formula from word descriptions
Q2 -5: creating sequences (tables of values) from recursive formula in the form Tn=...
Q6: identifying AP's, GP's or neither from sequences
Q7-9: creating sequences (tables of values) from recursive formula in the form Tn+1=...
Q10: creating sequences (tables of values) from recursive formula in the form Tn-1=...
Q11-12,15,16,17: creating sequences from multiple previous terms
Q13: recursive formula using the term counter(n) in the formula
Q14: finding unknowns in recursive formula
Q18-22: Growth and decay problems

My recollection of when we did this in Discrete was that these topics were covered over multiple chapters. When attempting 8A students faced difficulties in that the calculator has some limitations. I can usually maintain 1 chapter per lesson but in this case I let it run over three lessons and found some extra resources to supplement the topic as it left many students scratching their heads. This was hard as it chewed into the revision time I had left for exams.

Things to remember for next year:
1. Present multiple examples of recursive formula for the same sequence for Tn, Tn+1 and Tn-1.
2. Introduce the limitation that the calculator (in sequence mode) can only use up to two previous terms in its definition (eg. Tn+2=Tn+1+ Tn not Tn+3=Tn+2+Tn+1+Tn). We wasted a lot of time on this.
3. You cannot move freely between Tn-1, Tn, Tn+1 and Tn+2 representations if n itself is used in the formula. eg. Tn+1=Tn +3 is equivalent to Tn=Tn-1 +3 but Tn+1=Tn + n is not equivalent to Tn = Tn-1 + n
4. Be careful with the position when dealing with growth and decay. It is usually much easier to define T0 (Tzero) and Tn+1=.. as the initial value and formula. Thus when solving for n, n is the answer rather than n-1 (which caused no end of confusion amongst students).
5. Make students do the examples without a calculator unless it states otherwise. A lot of time can be wasted trying to make Sequence mode do things it is not intended to do.

Sequence mode and Calculator usage (What not to do).

Most sequences can be done in Sequence mode. Some cannot. Here's how to get into Sequence mode.

Let's put in Tn = 2Tn-1 -5 where T1=3 (q.11 from 3a MAT). We are looking for T1 to T5. Press Type in the menu.
Figure 1.

You will notice that the notation to the text is different in that "an" is used instead of Tn. Ignoring that, you will also notice that there is no option for "an", only for "an+1" or "an+2". (Blogger can't do subscripts so just put them in where needed!).

In this case it is not such a problem, we can just transpose our equation to Tn+1 = 2Tn + 5 as n itself is not used in the formula; The given value T1=3 now becomes T2=3, remembering that we are looking for T2 to T6 now (which is really T1-T5 of the original formula).

So now we put in T2.. Easy no? NO! If you look at Figure 1 we have options for Tn+1 where we are given a0 (the zero term) or a1(the first term). No option for a2.

The Saddler text does a pretty good job of making the calculator look clumsy and painful to use compared to paper and pen.

Here is a link to other CAS calculator posts.

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