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Odd Number Theorem

The sum of the first $n$ Odd Numbers is a Square Number:

$\displaystyle \sum_{k=1}^n (2k-1)$ $\textstyle =$ $\displaystyle 2\sum_{k=1}^n k-\sum_{k=1}^n 1 = 2 \left[{n(n+1)\over 2}\right]-n$  
  $\textstyle =$ $\displaystyle n(n+1)-n=n^2.$  

See also Nicomachus's Theorem




© 1996-9 Eric W. Weisstein
1999-05-26