Odd Number Theorem/Corollary
Jump to navigation
Jump to search
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 |
$\blacksquare$