# Natural Numbers and Induction (Analysis with and Introduction to ProofC)

1. Jan 22, 2007

### k3N70n

1. The problem statement, all variables and given/known data

Prove that: 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 for all n where n is a natural number

2. Relevant equations

Proof by induction:
a) p(1) is true
b) assume p(k) is true then prove p(k+1) for it

3. The attempt at a solution

I gave this a try for a while but I'm quite stuck.

So far using proof by induction:
p(1) = 1^3 = 1^2 is true

assume that p(k) = 1^3 + 2^3 + ... + k^3 = (1 + 2 + ... + k)^2 is true
then:
p(k+1) = 1^3 + 2^3 + ... + k^3 + (k + 1)^3
= (1 + 2 + ... + k)^2 + (k + 1)^3

Now I'm really not sure where to go or if I'm going the right way. Any sort of hint would be greatly appreciated. Thanks.

-Kentt

2. Jan 22, 2007

### IMDerek

Simplify 1+2+3+...k.

3. Jan 23, 2007

### Gib Z

$$\sum_{n=1}^k n= \frac{(k+1)(k)}{2}$$ btw.
You may also like to know that $$\sum_{n=1}^k n^3 = \frac{(k+1)^2k^2}{4}=(\frac{(k+1)k}{2})^2$$...o nvm Thats what your trying to prove...

4. Jan 23, 2007

### k3N70n

Thanks IMDerek. Got it done.