# Formula for Odd Numbers

Before going to learn the formula for odd numbers, let us recall what are odd numbers. Odd numbers are whole numbers that cannot be exactly divided by 2. For example, numbers 1, 3, 5, 7, 9,.. are odd numbers. The odd numbers formula helps in representing an odd number in general. Let us learn about the formula for odd numbers with a few examples in the end.

## What Is the Formula for Odd Numbers?

An odd number can be always expressed as 1 plus or minus an even number. We know that an even number is of the form 2n, where n is a whole number. Thus, the formula for odd numbers can be given as:

Formula for odd numbers = 2n +1

where n ∈* W *(whole numbers)

Another trick to check whether the number is odd or not we can directly divide it by 2. If the number is exactly divisible it is an even number, else it is an odd number.

## Sum of Odd Numbers Formula

We just read about a general formula to represent an odd number, we can also find the formulae to find the sum of first n odd numbers, the sum of their squares or cubes as follows:

- Sum of first n odd numbers = n
^{2}where n is a natural number. To calculate the sum of first n odd numbers without actually adding them individually. i.e., 1 + 3+ 5 +...........n terms = n^{2} - Sum of squares of first n odd numbers1
^{2}+ 3^{2}+ 5^{2}+...........(2n-1)^{2}terms = n(4n^{2 }-1)/3 - Sum of cubes of first n odd numbers 1
^{3}+ 3^{3}+ 5^{3}+...........(2n-1)^{3}= n^{2 }(2n^{2 }-1)

Let us see the applications of the formula for odd numbers in the following section.

## Examples on Odd Numbers Formula

**Example 1:** Check if 23 is an odd number using the formula for odd numbers.

**Solution:**

To find: Check if 23 is an odd number.

On dividing 23 by 2, it leaves the remainder as 1.

Thus, we conclude that 23 is an odd number.

**Answer: **23 is an odd number.

**Example 2:** Using the odd number formula solve the following.

a) 1+ 3 + 5 + 7 + ...................40 terms Find the sum of 40 terms.

b) Find the sum of odd numbers from 1 to 51.

**Solution: **

a) n = 40. so sum = n^{2} = 40^{2 }= 1600

b) Sum = [(1+51)/2]^{2 }= [52/2]^{2} =26^{2 }= 676

**Example 3:** The sum of 3 consecutive odd numbers is 21. Find the 3 numbers using odd numbers formula.

**Solution:** Let the 3 consecutive odd numbers be 2n+1, 2n+3 and 2n +5

given: (2n+1) + (2n+3) + (2n +5) = 21

6n + 9 = 21

6n = 12

n = 2

Thus 2n + 1 = 5

2n + 3 = 7

2n + 5 = 9

Therefore, the 3 consecutive odd numbers are 5, 7 and 9.

## FAQs on Odd Number Formula

### How to Determine the Odd Number Series Using Odd number Formula?

To determine the odd number series we will use the odd numbers formula which can be expressed as:

Odd Numbers = 2n + 1 where n ∈ W (whole numbers)

### What Are the Ways of Identifying Odd Numbers?

To identify an odd number we can directly divide it by 2. If the number is exactly divisible by 2 it is not an odd number. For example, 4 is not an odd number as it is exactly divisible by 2.

### What Does n Represent in Odd Number Formula (2n+1)?

To find the series of odd numbers we use the general odd number formula (2n+1). Here n represents the whole numbers. For identifying sum on n odd numbers we use formula n^{2} here n is a natural number.

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