# Find an easy way to evaluate the following series: 3+5+7+cdots+ 17+19+21?

Apr 17, 2016

$120$

#### Explanation:

Note that as you work inwards, groups of $24$ form.

Add the first and last numbers, $3$ and $21$, to get $24$.

Since the next two numbers inwards are just $2$ less and greater than the numbers we added before, their sum will also be $24$. That is:

$5 + 19 = \left(3 + 2\right) + \left(21 - 2\right) = \left(3 + 21\right) + \left(2 - 2\right) = 24$

The pattern continues:

$\left\{\begin{matrix}3 + 21 = 24 \\ 5 + 19 = 24 \\ 7 + 17 = 24 \\ 9 + 15 = 24 \\ 11 + 13 = 24\end{matrix}\right.$

Thus, the sum is $5 \times 24 = 120$.