Proof by Induction - $(\sum_{i=1}^{n}i)^2 = \sum_{i=1}^{n}i^3$

Asked Oct 30 '17 at 20:55

Active Oct 30 '17 at 20:55

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Hello Mathematics Community, currently I am trying to prove:

$(\sum_{i=1}^{n+1}i)^2 = \sum_{i=1}^{n}i^3$

by Induction.

My Progress:

Assumption: $(\sum_{i=1}^{n+1}i)^2 = \sum_{i=1}^{n}i^3$

Induction Hypothesis: $=(\sum_{i=1}^{n+1}i)^2 = \sum_{i=1}^{n+1}i^3$

Induction Step:

$(\sum_{i=1}^{n}i)^2 = \sum_{i=1}^{n}i^3$

$=(\sum_{i=1}^{n+1}i)^2 = \sum_{i=1}^{n+1}i^3$

$=(\sum_{i=1}^{n} i + \sum_{i=n+1}^{n+1} i)^2$

But how can I use my Assumption here ?

It is important that I want to prove this with the summation symbol and not with the closed-form equations.

sincerely, M.Hisoka

asked Oct 30 '17 at 20:55

M.Hisoka

I think you have a typo in the induction hypothesis – Daniel Ordoñez Oct 30 '17 at 20:59
1

I don't think you can prove this without using $\sum_{i=1}^ni=n(n+1)/2$, unless you intentionally avoid that. – Eclipse Sun Oct 30 '17 at 21:00
@EclipseSun so I am forced to use the closed-form equations ? – M.Hisoka Oct 30 '17 at 21:01
Your hypothesis contradicts what you are trying to prove. There is something wrong with your indexes. Also note that you can't use your assumption because you are assuming that what you are trying to prove is true! You can't use such an assumption – Francisco José Letterio Oct 30 '17 at 21:04

0 Answers0