Odd Number Theorem/Corollary

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Theorem

A recurrence relation for the square numbers is:

$S_n = S_{n - 1} + 2 n - 1$


Proof

\(\ds S_n\) \(=\) \(\ds \sum_{j \mathop = 1}^n \paren {2 j - 1}\) Odd Number Theorem
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^{n - 1} \paren {2 j - 1} + \paren {2 n - 1}\) Definition of Summation
\(\ds \) \(=\) \(\ds S_{n - 1} + \paren {2 n - 1}\) Odd Number Theorem

$\blacksquare$