How do you find the sum of all odd numbers?

You can find the sum of a series of odd numbers using a couple of effective methods.

Methods for Calculating the Sum of Odd Numbers

There are primarily two ways to find the sum of odd numbers, depending on whether you're dealing with a specific sequence or a general principle.

1. Using the Formula for a Specific Sequence

If you know the first and last odd numbers in a sequence and the number of odd numbers in that sequence, you can use the following formula:

• Sn = n/2 × (a + l)

  • Where:
    • Sn = Sum of the odd numbers
    • n = Number of odd numbers in the sequence
    • a = First odd number in the sequence
    • l = Last odd number in the sequence

Example: Find the sum of the first 5 odd numbers (1, 3, 5, 7, 9).

• n = 5

• a = 1

• l = 9

  Sn = 5/2 × (1 + 9) = 5/2 × 10 = 25

2. Using the Simplified Formula for the Sum of the First 'n' Odd Numbers

A shortcut exists for finding the sum of the first 'n' odd numbers. The sum is simply the square of 'n'.

• Sn = n²

  • Where:
    • Sn = Sum of the first 'n' odd numbers
    • n = The number of odd numbers you are summing

Example: Find the sum of the first 5 odd numbers.

• n = 5

  Sn = 5² = 25

Choosing the Right Method

• Use Sn = n/2 × (a + l) when you have a specific sequence of odd numbers and know the first, last, and total count.
• Use Sn = n² when you want to find the sum of the first 'n' odd numbers (starting from 1).

In summary, the method you choose depends on the information you're given about the sequence of odd numbers.