1. ## Mathematical Induction

Hi there,

I'm totally dumbfounded by this question. Could anyone give me a basis on where to start so I can at least try to work it out myself?

Use mathematical induction to show that:

$\displaystyle S(n) = 3*2^{n-1} -2$
is the solution for the reccurence relation:
$\displaystyle T(n)=2T(n-1)+2$ for $\displaystyle n > 1$ and $\displaystyle T(1) = 1$

2. To prove $\displaystyle T(n)=S(n)$ for all positive integers $\displaystyle n$ all you have to do is show

(i) that $\displaystyle T(1)=S(1)$, and
(ii) that if $\displaystyle T(k-1)=S(k-1)$ for some $\displaystyle k>1$ then also $\displaystyle T(k)=S(k)$ for that very same $\displaystyle k$.

Here we go.

Let $\displaystyle S(n)=3\times2^{n-1}-2$.

Initial step: $\displaystyle S(1)=3\times2^0-2=3-2=1$, so $\displaystyle T(1)=S(1)$.

Inductive step: Assume that $\displaystyle T(k-1)=S(k-1)$ for some $\displaystyle k>1$.

Then $\displaystyle T(k)=2T(k-1)+2=2S(k-1)+2=2(3\times2^{k-2}-2)+2=3\times2^{k-1}-2=S(k)$.

Thus, by the principle of induction, $\displaystyle T(n)=S(n)$ for all positive integers $\displaystyle n$.

3. Thanks, that helped a lot. I have a follow up question here.

If 1 is added to the recurrence relation such that:

$\displaystyle T(n) = 2T(n-1) + 3$ for $\displaystyle n > 1$ and $\displaystyle T(1) = 0$

What is the new equation for S(n)?
Prove it by induction.
Am I basically reversing the steps in the first answer to come up with a new equation for S(n)? This is what I've come up with:

$\displaystyle T(k) = 2T(k-1)+3=2S(k-1)+3=3(3*2^{k-2}-2)+3=3x2^{k-1}-3=S(k)$