# The Trick for Quickly Calculating the Sum of First N Odd [[Integer Numbers]] $ \sum_{a=0}^{n-1}(2a+1) = n^2 $ This formula can be verified visually using the following picture which shows the sum of first 5 odd numbers (in alternating gray colors from bottom to top): $1+3+5+7+9 = 25$ ![[fPDO16cpCQ-group-21.svg.png|300]] The sequence of numbers formed by this formula is known as the Square Numbers Series Sequence (1, 4, 9, 16, 25,...). It is an example of an [[Arithmetic Series]]. For more info see this Brilliant quiz: [Odd Square Sums](https://brilliant.org/practice/odd-square-sums/) For the trick to calculate the sum of first n even numbers see [[The Trick for Quickly Calculating the Sum of First N Even Integers]]. ## Proof Proof is simply applying the formula for the arithmetic series ![[Arithmetic Series#^3b0qf5]] $ \begin{align} \sum_{a=0}^{n-1}(2a+1) &= \frac{n}{2}(2a_1+d(n-1)) \\ &=\frac{n}{2}(2+2(n-1)) \\ &=\frac{n}{2}(2+2n-2) \\ &=n^2 \end{align} $