Induction proof (solution included but still confused).

[s3a]

Homework Statement

The problem (and its solution) is attached as Problem.jpg.

Homework Equations

Base case, induction hypothesis, and induction step.

The Attempt at a Solution

I see that the base case is correct and I also see how everything under "Proof:" is done algebraically. As I was writing this, I think I unconfused myself for some things but something must be wrong with either the way I'm interpreting this or the way the problem is because let k = 5: T(2^5) = 32 ≠ 3^(5+1) - 2^(5+1) = 3^6 - 2^6 = 665. Assuming I am wrong, what am I not seeing? Edit: I'm thinking it has something to do with the n > 1 in the definition of T(n) but I can't pin down the issue specifically.

Any input would be greatly appreciated!
Thanks in advance!

 

[Max.Planck]

The equation is T(2^5) = 3^5-2^6 not 2^5 = 3^6-2^6 :).

 

[s3a]

Accidentally triple posted.

 

[s3a]

Accidentally triple posted.

 

[s3a]

Oh, ya! :)

I still have another problem and it might be a similar one but I feel I'm getting close:

Let k = 1, T(1) = 1, T(2^1) = T(2) = 3*T(1) + 1 = 4

but T(2^1) = T(2) = 4 ≠ 3^2 - 2^2 = 5.

What am I doing wrong now?

Edit: Found my mistake. the +n part in the T(n) for n > 1 definition, I did +1 instead of +2 because I was doing it in my head.

Thanks for the help!