Question

1*2 + 2*2^2 + 3*2^2 + ...+n*n^2= (n-1)2^n+1 + 2
use induction only final answer

Answer 1

Answer:

For n > 1, 2 + 22 + 23 + 24 + ... + 2n = 2n+1 – 2

Let n = 1. Then:

2 + 22 + 23 + 24 + ... + 2n = 21 = 2

...and:

2n+1 – 2 = 21+1 – 2 = 22 – 2 = 4 – 2 = 2

So (*) works for n = 1.

Assume, for n = k, that (*) holds; that is, that

2 + 22 + 23 + 24 + ... + 2k = 2k+1 – 2

Let n = k + 1.

2 + 22 + 23 + 24 + ... + 2k + 2k+1

= [2 + 22 + 23 + 24 + ... + 2k] + 2k+1

= [2k+1 – 2] + 2k+1

= 2×2k+1 – 2

= 21×2k+1 – 2

= 2k+1+1 – 2

= 2(k+1)+1 – 2

Then (*) works for n = k + 1.

Note this common technique: In the "n = k + 1" step,