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The sum of the first [math]n[/math] odd natural number is [math]n^2[/math], but what is the sum of first [math]n[/math] natural even number?
How is the formula for the sum of even numbers derived?
The formula is derived by adding up the first [math]n[/math] even numbers which can be represented as [math]2+4+6+\cdots + 2n[/math]. Factoring out 2 gives [math]2(1+2+3+\cdots +n)[/math], which simplifies to [math]2\left(\frac{n(n+1)}{2}\right)[/math] and finally [math]n(n+1)[/math].