Problem 44 (difficulty: 3/10)

Prove that the following identity holds for all positive integer \(\displaystyle n\):

\(\displaystyle 1^3 +\ldots +n^3 =\left( \frac{n\cdot (n+1)}{2} \right) ^2 .\)

Solution:

Induction on \(\displaystyle n\). For \(\displaystyle n=1\) both sides equal to \(\displaystyle 1\). If the statement holds for \(\displaystyle n\), then for \(\displaystyle n+1\) we have 1

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