• Support PF! Buy your school textbooks, materials and every day products Here!

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

  • Thread starter k3N70n
  • Start date
67
0
1. Homework Statement

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

2. Homework 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
 

Answers and Replies

19
0
Simplify 1+2+3+...k.
 
Gib Z
Homework Helper
3,344
4
[tex]\sum_{n=1}^k n= \frac{(k+1)(k)}{2}[/tex] btw.
You may also like to know that [tex]\sum_{n=1}^k n^3 = \frac{(k+1)^2k^2}{4}=(\frac{(k+1)k}{2})^2[/tex]...o nvm Thats what your trying to prove...
 
67
0
Thanks IMDerek. Got it done.
 

Related Threads for: Natural Numbers and Induction (Analysis with and Introduction to ProofC)

Replies
5
Views
1K
Replies
1
Views
646
Replies
4
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
Replies
6
Views
4K
Replies
22
Views
3K
Replies
5
Views
3K
Top