Mathematical Induction

**GreenDay14** · December 3rd 2009, 02:11 PM

Hey, i have a question here that reads:

Prove by mathematical induction that 1³ + 1² + ... + n³ = (n²(n+1)²)/4 for any natural number n.

Could anyone get me started on this? Any help would be greatly appreciated. Thanks.

**artvandalay11** · December 3rd 2009, 02:15 PM

Originally Posted by GreenDay14

Hey, i have a question here that reads:

Prove by mathematical induction that 1³ + 1² + ... + n³ = (n²(n+1)²)/4 for any natural number n.

Could anyone get me started on this? Any help would be greatly appreciated. Thanks.

Don't you mean $1^3+2^3+3^3+...+n^3$

**GreenDay14** · December 3rd 2009, 02:25 PM

Nope, reading it straight out of the book.

**artvandalay11** · December 3rd 2009, 03:01 PM

the expression $1^3+1^2+...n^3$ doesnt make any sense

I will assume the book/you actually want to prove $\sum_{k=1}^n k^3=\frac{n^2(n+1)^2}{4}$ which is true, can you prove this now that I have stated the actual problem or do you still need help

**emakarov** · December 3rd 2009, 04:22 PM

You can look at a similar thread here.

**guildmage** · December 3rd 2009, 06:59 PM

Could you verify the first three steps? For n = 1, n = 2, and n = 3.

**Roam** · December 5th 2009, 10:00 AM

I think there might be a typo because in this problem you can't prove the base case for your proposition:

$P(1) = \frac{1^2(n+1)^2}{4} = 1 \neq (1^3 + 1^2 + 1) = 3$

If we can't prove the base case, we can't move onto the inductive step.

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