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