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I recently bought the the Dover Series book on Number Theory, and the 2nd example on page 5 asks your to prove

[tex]1^3 + 2^3 + 3^3 ... + n^3 = (1 + 2 + 3....)^2 [/tex]

Now, we've already proved that [tex]S_n = \frac{n(n+1)}{2}[/tex]

So here's how I proved it...

[tex]

(S_n)^2 = (\frac{n(n+1)}{2})^2 [/tex]

before we proved how [tex] S_k+1 = S_k + (k + 1) [/tex]

Which lead me to...

[tex] \frac{1}(k+1)^2((k+1)+1)^2{4} + (k+1)^2 [/tex]

[tex] = \sqrt{\frac{(k+1)^2((k+1)+1)^2}{4} + (k+1)^2} [/tex]

[tex] = \frac{(k+1)((k+1)+1)}{2} + (k+1) [/tex]

therefor...

[tex] S_k+1 = Sk + (k+1) [/tex]

Now I'm worried that I didnt really solve anything. I'm totally knew at this, and I'm open to criticism and help, just be kind :)

(ps.. this is my first time posting formulas, hopefully I did it right)

Thanks

dleacock

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# First attempt at a proof

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