Cube Number as Difference between Squares of Triangular Numbers

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Theorem

Let $n \in \Z_{> 0}$ be a positive integer.

Then:

$n^3 = {T_n}^2 - {T_{n - 1} }^2$

where $T_n$ denotes the $n$th triangular number.


Proof

\(\displaystyle {T_n}^2 - {T_{n - 1} }^2\) \(=\) \(\displaystyle \paren {\frac {k \paren {k + 1} } 2}^2 - \paren {\frac {\paren {k - 1} k} 2}^2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {k^2 + k}^2 - \paren {k^2 - k}^2} 4\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {k^4 + 2 k^3 + k^2} - \paren {k^4 - 2 k^3 + k^2} } 4\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {4 k^3} 4\)
\(\displaystyle \) \(=\) \(\displaystyle k^3\)

$\blacksquare$


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