Recursive Series

Question

Am I doing this right? What's a good way to think about the different f(n)'s. f(n + 1) is the "next" element in the series? f(n) is the current element? The previous element?

Find $f(1), f(3), f(4),$ and $f(5)$ if $f(n)$ is defined recursively by $f(0) = 3$ and for $n = 0,1,2...$

[a.] $f(n+1) = -2f(n)$

\begin{align*} \text{Basis Step: } f(0) = 3& \\ \\ f(0 + 1) = -2(3) = -6 \\ f(1 + 1) = -2(-6) = 12 \\ f(2 + 1) = -2(12) = -24 \\ f(3 + 1) = -2(-24) = 48 \\ f(4 + 1) = -2(48) = -96 \end{align*}$

Answer 1

The computations are correct. Yes, $f(n)$ can be thought of as the current element of the sequence, and $f(n+1)$ as the next element. The recurrence (in this case) tells you how to compute the next element, if you know the current one.

Answer 2

A clearer way for me to see the recurrence relation is by rewriting it as

$f(0) = 0$

$f(n) = -2f(n-1)$

Looking at it this way says whatever term in the sequence you are interested in is twice the previous term.