## Sum of n odd numbers

The sum of the first n odd natural numbers is a basic mathematical problem that is used in many mathematical fields such as combinatorics, number theory, and algebra. It is a common exercise in primary and secondary education and is also used in various mathematical competitions.

Formula:

The sum of the first n odd numbers can be expressed by the formula:

1 + 3 + 5 + 7 + … + (2n – 1) = n^2

where n is a positive integer.

Proof:

To prove the formula, we will use mathematical induction.

Base Case:

For n = 1, the sum of the first n odd numbers is:

1 = 1^2

which is true.

Induction Hypothesis:

Assume that the formula holds for some positive integer k, i.e.

1 + 3 + 5 + 7 + … + (2k – 1) = k^2

Induction Step:

We will now show that the formula also holds for k+1.

1 + 3 + 5 + 7 + … + (2k – 1) + (2(k+1) – 1)

= k^2 + (2(k+1) – 1)

= k^2 + 2k + 1

= (k+1)^2

Therefore, the formula holds for all positive integers n by mathematical induction.

Example:

Let’s find the sum of the first 5 odd natural numbers using the formula:

1 + 3 + 5 + 7 + 9 = 5^2 = 25

Therefore, the sum of the first 5 odd natural numbers is 25.

Applications:

The sum of the first n odd natural numbers is used in various mathematical applications such as calculating the number of odd divisors of a given number, finding the sum of the first n positive odd integers in a square, and in the analysis of combinatorial problems. It is also used in physics and engineering to calculate the gravitational force between two masses, the energy of a photon, and in the analysis of electric fields.

here are some examples of the sum of n odd numbers:

- The sum of the first 5 odd numbers is 1 + 3 + 5 + 7 + 9 = 25.
- The sum of the first 10 odd numbers is 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100.
- The sum of the first 15 odd numbers is 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 = 225.
- The sum of the first 20 odd numbers is 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 + 33 + 35 + 37 + 39 = 400.

In this post, we have discussed the formula for the sum of the first n odd natural numbers and proved it using mathematical induction. We have also provided an example of how to use the formula and discussed some of its applications. The sum of the first n odd natural numbers is a fundamental mathematical problem that is used in many mathematical fields and has various applications in science and engineering.