# What is the sum of the first 500 odd numbers?

Mar 25, 2018

250000

#### Explanation:

The first one is 1, the last is $2 \times 500 - 1 = 999$. Their average is 500. Since the numbers are in an AP, the average of all 500n of them is also the same, namely 500. Thus, the sum is

$500 \times 500 = 250000$

In general, the sum of the first $n$ odd numbers is

$n \times \frac{1}{2} \left(1 + \left(2 n - 1\right)\right) = {n}^{2}$