Find the sum of series 1, 3, 5, 7, 9,…… up to n terms.

To find the sum of the series 1, 3, 5, 7, 9, and so on, up to n terms, we first need to recognize that this series consists of the first n odd numbers. The pattern is quite simple: each term increases by 2 from the previous term. Let's break down how to calculate the sum effectively.

Understanding the Series

The series of odd numbers can be expressed as:

• 1st term: 1
• 2nd term: 3
• 3rd term: 5
• 4th term: 7
• 5th term: 9
• ... and so on.

Identifying the Formula

There is a well-known formula for the sum of the first n odd numbers. The sum can be calculated using the formula:

Sum = n²

This means that if you want to find the sum of the first n odd numbers, you simply square n. For example:

• If n = 1, the sum is 1² = 1.
• If n = 2, the sum is 2² = 4 (1 + 3).
• If n = 3, the sum is 3² = 9 (1 + 3 + 5).
• If n = 4, the sum is 4² = 16 (1 + 3 + 5 + 7).

Why Does This Work?

This formula works because of the properties of odd numbers. Each odd number can be thought of as a step in a sequence that builds upon the previous sums. When you add the first n odd numbers, you are essentially creating a perfect square. For instance, if you visualize the sum of the first three odd numbers (1, 3, and 5), you can arrange them in a square formation:

• 1 (1x1 square)
• 1 + 3 (2x2 square)
• 1 + 3 + 5 (3x3 square)

Example Calculation

Let’s say you want to find the sum of the first 5 odd numbers:

• 1 + 3 + 5 + 7 + 9 = 25

Using the formula, we can also calculate this:

Sum = 5² = 25

Final Thoughts

In summary, the sum of the series of the first n odd numbers is simply n squared. This elegant relationship not only simplifies calculations but also highlights the beauty of number patterns in mathematics. So, whenever you need to find the sum of the first n odd numbers, just remember to square n!