I am to use mathematical induction to prove that:
$$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2 $$
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Adriano
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4Great! Then I think you should start doing it, now that we know your intentions! – davidlowryduda Aug 22 '14 at 18:00
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Do you what Gauss did with the problem $1+\cdots+100$? XD – Troy Woo Aug 22 '14 at 18:00
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Base case: if $n=1$ then $1^3=1^2$.
Induction step: you know that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}$$ So: $$1^3+\cdots+n^3+(n+1)^3=(\frac{n(n+1)}{2})^2+(n+1)^3$$ So all what remains is to prove that: $$(\frac{n(n+1)}{2})^2+(n+1)^3=(\frac{(n+1)(n+2)}{2})^2$$
Math Gems
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