Cube as Difference between Two Squares

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Theorem

A cube number can be expressed as the difference between two squares.


Proof

\(\displaystyle n^3\) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n k^3 - \sum_{k \mathop = 1}^{n - 1} k^3\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\frac {n^2 \paren {n + 1}^2} 4} - \paren {\frac {\paren {n - 1}^2 n^2} 4}\) Sum of Sequence of Cubes
\(\displaystyle \) \(=\) \(\displaystyle \frac {n^2 \paren {\paren {n + 1}^2 - \paren {n - 1}^2} } 4\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\frac {n \paren {n + 1} } 2}^2 - \paren {\frac {n \paren {n - 1} } 2}^2\)

$\blacksquare$


Sources