How to Find the Sum of Odd Integers?

The simplest way to find the sum of the first n odd integers is to square n: Sum = n².

Let's explore this with a few methods and explanations:

Understanding the Concept

Odd integers are whole numbers that are not divisible by 2 (e.g., 1, 3, 5, 7, ...). Finding the sum of a sequence of odd integers can be approached in a few ways.

Method 1: The Formula

As mentioned above, the sum of the first n odd integers is simply n².

• Example 1: The sum of the first 3 odd integers (1 + 3 + 5) is 3² = 9.
• Example 2: The sum of the first 5 odd integers (1 + 3 + 5 + 7 + 9) is 5² = 25.

Method 2: Arithmetic Progression

Odd integers form an arithmetic progression where the common difference is 2. We can use the formula for the sum of an arithmetic series:

S_n = (n/2) * [2a + (n - 1)d]

Where:

• S_n is the sum of the first n terms
• n is the number of terms
• a is the first term (which is 1 for odd integers)
• d is the common difference (which is 2 for odd integers)

Let's simplify this formula for the case of summing odd integers:

S_n = (n/2) [2(1) + (n - 1)2]
S_n = (n/2) [2 + 2n - 2]
S_n = (n/2) * [2n]
S_n = n²

As you can see, this formula also simplifies to n².

• Example: Find the sum of the first 4 odd integers.

  • n = 4, a = 1, d = 2
  • S₄ = (4/2) * [2(1) + (4 - 1)2]
  • S₄ = 2 * [2 + 6]
  • S₄ = 2 * 8 = 16 (which is also 4²)

Method 3: Iterative Summation

For a small number of odd integers, you can simply add them up directly.

• Example: Find the sum of the first 2 odd integers. 1 + 3 = 4 (which is also 2²).

When the Sequence Doesn't Start at 1

If you need to find the sum of a series of consecutive odd integers that doesn't start with 1, you need to adjust your approach. For example, find the sum of odd integers from 5 to 11 (5+7+9+11).

1. Find the sum of all odd integers up to the last number in your series (in this case, 11). To do that, you need to figure out how many odd integers there are from 1 to 11. They are: 1, 3, 5, 7, 9, 11. That's six odd integers. So the sum of the first 6 odd integers is 6² = 36.
2. Find the sum of all odd integers before your series starts (in this case, before 5). That's just 1 and 3. That's two odd integers. So the sum of the first 2 odd integers is 2² = 4.
3. Subtract the second sum from the first sum. 36 - 4 = 32. Therefore, 5 + 7 + 9 + 11 = 32.

In Summary

Finding the sum of the first n odd integers is straightforward: simply square n. For more complex sequences, apply the arithmetic progression formula or adjust the n² formula accordingly, as demonstrated above.