The sum of a given sequence of numbers is known as a series, and many series  both infinite and finite  have known sums. For example, a mathematician named Carl Gauss is famous for determining a formula for the first N consecutive numbers as a boy in the 18th century. Using a variation of Gauss' result, you will find a simple expression for the sum of consecutive odd numbers.

Determine the number (N) of consecutive odd numbers you're adding. If your series is given in sigma notation, this is the finishing index (on top of the sigma) minus the starting index (beneath the sigma) plus one. Alternatively, subtract the largest odd number in your series from the smallest, divide this difference by two and add one. For example, if adding the odd numbers from 7 to 45, N = (45  7) / 2 + 1 = 20.

Multiply the smallest number in the series by the number of numbers in the series you determined in Step 1. For example, if adding the odd numbers from 7 to 45, multiply 7 by N (20) = 140.

Multiply N by N  1, and add this to the product you found in Step 2. For example, if adding odd numbers from 7 to 45, where N = 20, add the product N (N  1) = 20 19 = 380 to 7 20 = 140 to get 520. In other words, the formula is: min N + N * (N  1), where N is the number of consecutive odd numbers to sum, and "min" is the smallest of these.
Tips & Warnings
 This formula also works for consecutive even numbers.
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