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Step 1: Show that the statement holds for the basis case $n = 1$

\begin{align} \sum_{i=1}^1 i^3 & = (\sum_{i=1}^1)^2 \\ \Leftrightarrow 1^3 & = 1^2 \\ \Leftrightarrow 1 & = 1 \\ \end{align}

Step 2: Assume that $n$ holds.

Step 3: Show that on the assumption that the summation is true for n, it follows that it is true for n + 1.

\begin{align} \sum_{i=1}^{n+1} i^3 & = (\sum_{i=1}^{n+1} i)^2 \\ \Leftrightarrow \sum_{i=1}^{n} i^3 + (n+1)^3 & = (\sum_{i=1}^{n} i + (n+1))^2 \\ & = (\sum_{i=1}^n i)^2 + 2(\sum_{i=1}^n i)(n+1) + (n+1)^2 \\ & = (\sum_{i=1}^n i)^2 + 2(\frac{n(n+1)}{2})(n+1) + (n+1)^2 \\ & = (\sum_{i=1}^n i)^2 + n(n+1)^2 + (n+1)^2 \\ & = (\sum_{i=1}^n i)^2 + (n+1) \cdot (n+1)^2 \\ \Leftrightarrow \sum_{i=1}^{n} i^3 + (n+1)^3 & = (\sum_{i=1}^n i)^2 + (n+1)^3 \\ \Leftrightarrow \sum_{i=1}^{n} i^3 & = (\sum_{i=1}^n i)^2 \\ \end{align}

This is the case of n assumed to be true in step 2, so n + 1 holds as well.

Above the arithmetic series was used: $\sum_{i=1}^n i = \frac{n(n+1)}{2}$

QED