# The Trick for Quickly Calculating the Sum of First N Even [[Integer Numbers]] Closely follows the principle used behind [[The Trick for Quickly Calculating the Sum of First N Odd Integers]]. $ \sum_{a=1}^n {2a} = n(n+1) $ Here is a picture visually explaining the formula. It shows the sum of first 4 even numbers (alternating blue and pink): $2 + 4 + 6 + 8 = 4 \cdot 5 = 20$ ![[Screenshot 2021-02-15 at 16.34.59.png]] As the sequence of numbers formed by the formula for the sum of first n odd number is known as the Square Numbers Series Sequence, the one formed by the formula for the sum of the first n even numbers could be be called the *Rectangle Numbers Series Sequence* (this extrapolation is my own and does not come from Brilliant or elsewhere). Both are examples of an [[Arithmetic Series]]. For more info see this Brilliant quiz: [Odd Square Sums](https://brilliant.org/practice/odd-square-sums/) ## Proof Proof is simply applying the formula for the arithmetic series ![[Arithmetic Series#^3b0qf5]] Note the different results, depending on whether $a_1$ is considered to be 0 or 2. In the illustration above, the first even number is 2. $ \begin{align} \sum_{a=1}^n {2a} &= \frac{n}{2}(2a_1+d(n-1)) \\ &=\frac{n}{2}(2a_1+2(n-1)) \\ &=\frac{n}{2}(2a_1+2n-2) \\ &=n(a_1+n-1) \end{align} $ $ \begin{array}{l|l} a_1 = 0 & a_1 = 2 \\ \hline = n(0+n-1) & = n(2+n-1) \\ =n(n-1) & =n(n+1) \end{array} $