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EDIT: Adding in more information that is hopefully useful. This is part of a multi step question I'm trying to answer for my homework.

First we were given a1 = -3 and a formula ak+1 = ak -1, for all integers k >= 1

Using this formula we then had to figure out the first seven numbers in the sequence it produced:

$a_1 = -3$

$a_2 = 4(-3)-1$

$a_3 = 4(4(-3)-1)-1 = 4_2 (-3) - 4(1)-1$

and so on and so forth, then using all that information we arrive at the part I'm stuck on below, which is finding an explicit formula for an using iteration

END EDIT

My textbook is of no help, and Google isn't being helpful either. I don't want an answer to my question, rather I want some help understanding how to properly guess an explicit formula using iteration. This is what I have so far:

$a_n = 4^n(-3) - 4^{n-1}\cdot 1 - 4^{n-2}\cdot 1 - \cdots- 4^3\cdot 1 - 4^2\cdot 1 - 4 \cdot 1 - 1$

I'm looking over my class notes but for some reason I forgot to explain a critical step in the process, that is reducing the latter half of the equation to help me come up with a guess for an explicit formula.

If someone could walk me through it I'd be appreciative. Emphasis on walk me through, I don't really want the answer, as I'd rather learn it myself.

EQJ
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Daniel Love Jr
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    Am I the only one who doesn't know what this expression is supposed to be? (If anyone has an idea and could edit it in, that would be fantastic) – MT_ Apr 25 '14 at 00:31
  • What do you mean by "what it is supposed to be?" We were given a formula, then had to find the first 7 numbers in the sequence that said formula produced, then were asked to guess an explicit formula based off of the sequence we figured out, which is what I posted above. – Daniel Love Jr Apr 25 '14 at 00:33
  • I couldn't tell what your expression is. I'm still kind of confused (why is every term being multiplied by $1$? Why is the first term being multiplied by $-3$ instead?) but you giving what the formula which produced these terms helps somewhat. (Though I think you meant $a_k = 4 a_{k-1} - 1$) – MT_ Apr 25 '14 at 00:47
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    You should learn latex... – EQJ Apr 25 '14 at 00:48
  • @MichaelT I just copied what I got from the professor, which is 4k+1 = 4ak - 1. For the multiplication thing, that's just how iteration is supposed to work? (to the best of my knowledge and notes) Might just be how it's being taught to me. – Daniel Love Jr Apr 25 '14 at 00:52

1 Answers1

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$$a_{n+1}=:b_n=-3\times4^n-\sum^{n-1}_{i=0}4^i\\ \implies \boxed{a_n=\dfrac{1}{3}\left(1-4^{n}\right)-3\times4^n}\\ \text{Can you see why?}$$

  • No, I am not understanding how to arrive at that conclusion with the information I have. – Daniel Love Jr Apr 25 '14 at 00:48
  • @DanielLoveJr Find the partial sum, perhaps by induction. –  Apr 25 '14 at 00:49
  • Well I was under the impression that I couldn't use induction until I had a formula with which to test via induction. The next step, after guessing a formula, is to test it with induction, I just don't know how to arrive at a proper guessed formula. – Daniel Love Jr Apr 25 '14 at 00:53
  • @DanielLoveJr Assume that $\sum^{n-1}_{i=0}4^i=\dfrac{1}{3}(4^n-1)$. Prove this by induction. Then, "work backwards". –  Apr 25 '14 at 00:55
  • No I am still lost, using the above formula I show a sub 1 as being equal to -13, when according to my previous calculations a sub 1 should be -3 and a sub 2 should be -13 – Daniel Love Jr Apr 25 '14 at 01:06
  • @DanielLoveJr I made a minor mistake - please see my edit. –  Apr 25 '14 at 01:08