## Introduction

Sum of Odd Natural Numbers Formula will give you a vivid description of this chapter on odd numbers. These solutions will widely help during the exams and also for its preparation. These solutions contain essential question answers and solvable questions that may come in the examination. These solved questions of the Sum of Odd Natural Numbers Formula will make revision easier for students before the exams. Sum of Odd Numbers Formula is an easy and scoring chapter. During the exams, if you follow these solved questions, it will clear all doubts. These solutions will help understand the chapter both critically and logically.

### Short Description of Sum of Odd Natural Numbers

The Sum of Odd Natural Numbers Formula is explained in simple steps here. These explanations are easy to understand and clear all doubts at once. The formula provides for easy exam preparation and completing homework. The solution provided will help to solve any questions related to this chapter. Sum of Odd Natural Numbers Formula will make the subject interesting and fun to learn. Sum of Odd Natural Numbers is a great chapter explaining the basics of graphs. It is an easy and scoring chapter. Preparing these solved questions will help with homework and exams as well.

### What is an Odd Number?

An odd number is a whole number that cannot give two equivalent resultants when divided by Two. some of the examples are 1, 3, 5, 7…..and so on.

An Odd number is an integer of the form n = 2k + 1 or alternately n= 2k − 1, Where n can be any natural number.

### General Equation For Odd Numbers

From the above heading, it is clear that the odd numbers are 1, 3, 5, 7, 9, 11, …and so on. Students can locate an Arithmetic Progression Sequence if look closely (AP), The AP can be integrated into the formula as follows:

The equation for odd numbers = 2n±1

Odd Number={2n+1) if n=0,1,2,3, and so on

(2n−1) if n=1,2,3, and so on.

Let the sum of first n odd numbers be Sn

S_{n} = 1+3+5+7+9+…………………..+(2n-1) ……. (1)

By Arithmetic Progression, we know, for any series, the sum of numbers is given by;

S_{n}=1/2×n(2a+(n-1)d) ……..(2)

Where,

n = number of digits in the series

a = First term of an A.P

d= Common difference in an A.P

Therefore, if we put the values in equation 2 with respect to equation 1, such as;

a=1 , d = 2

Let, last term, l = (2n-1)

S_{n} = ( n/2) × (a+l )

S_{n} = (n/2) × (1 + 2n – 1)

S_{n} = (n/2) × (2n) = n^{2}

Sum of n odd numbers = n^{2}

### Sum of 1 to 10 Odd Numbers

The table for the sum of odd numbers is shown below.

### Importance of Sum of Odd Natural Numbers Formula

Sum of Odd Natural Numbers Formula helps to prepare for exams without any worries. It will make students work much more manageably.

With the Sum of the Odd Natural Numbers Formula, a student can learn wisely and work hard.

Make a routine and allot time for different subjects. Routine makes learning fun and values time.

A key to good scores is to revise the previous year’s question paper from the Sum of Odd Natural Numbers.

It provides all the short answer type questions and Solved examples for the student guide.

### Solved Examples

1) Define the sum of odd numbers from 1 to infinity?

Solution: Using Arithmetic Progression, we can find the sum of odd numbers from 1 to infinity. As we know, the odd numbers are the numbers that are not divisible by 2. They are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 and so on. Now, we have to search for the sum of these numbers.

LetS_{n} be the sum of the first n odd numbers.

Or, S_{n} = 1 + 3 + 5 + 7 + 9 +…………………..+ (2n - 1) (Equation 1)

With the help of Arithmetic Progression(AP) here, we know, for any series, the sum of numbers is given by;

or, S_{n} = ½ × n(2a+(n−1)d)

2a+(n−1)d ……..(Equation 2)

Where,

n = number of digits present in the series

a = It is termed as First term of an A.P

d= it is the common difference in an A.P

Therefore, if we put the values in equation 2 with respect to equation 1, we will get

a = 1, d = 2

Let, last term be, l = (2n - 1)

or, S_{n} = (n/2) × (a + l)

or, S_{n} = (n/2) × (1 + 2n – 1)

or, S_{n} = (n/2) × (2n) = n^{2}

2) What is the sum of the odd numbers from count 1 to 50?

Solution: We already know that, from 1 to 50, there are 25 odd numbers present.

hence, n = 25

By the formula of the sum of odd numbers we come to know

or, S_{n} = n^{2}

or, S_{n} = 25^{2} = 625

3) What is the sum of odd numbers from 1 to 99?

Solution: We already know that, from 1 to 99, there are 50 odd numbers present.

hence, n = 50

By the formula of the sum of odd numbers we come to know

or, S_{n} = 50^{2}

or, S_{n} = 50^{2} = 2500

**1. Prove that Sum of ‘n’ Consecutive Odd Numbers = n ^{2}?**

a) We have to understand the pattern of odd numbers sequence in order to prove their sum. The total numbers of any set of the ordered odd numbers that are beginning with 1 are always equal to the square of their number of digits. Those are added together. If 1, 3, 5, 7, 9, 11, …, (2n - 1) are the odd numbers, then we can say

The sum of the first odd number is equal to 1.

Sum of the first two odd numbers is equal to 1 + 3 = 4 (4 = 2 x 2).

Sum of the first three odd numbers is equal to 1 + 3 + 5 = 9 (9 = 3 x 3).

Sum of the first four odd numbers is equal to 1 + 3 + 5 + 7 = 16 (16 = 4 x 4).

b) The number of digits that are added collectively in an ordered manner is always equal to the square root of the total number.

The sum of the first odd number is equal to 1.

The square root of 1, √1 = 1, so from here we understand that only one digit was added.

Sum of consecutive two odd numbers is equal to 1 + 8 = 9.

The square root of 9, √9 = 3, so here also we can see two digits were added.

Hence, from the above-proved sums, we can prove the formula. In order to find the sum of the first n-odd numbers is n x n or n^{2}.

For example, if we put n = 12 and then we have 12 x 12 = 144. This is equal to the sum of the first 12 odd numbers.

**2. What are the Benefits of Studying the Sum of Odd Numbers Formula?**

All chapters that are in the book are simple to build knowledge of math. Foundational concepts of Math are used interchangeably in many day-to-day applications, businesses, and jobs. Making complex calculations to simplify problems, learning Math changes our minds to think broadly, in a creative manner. During the exams, if you follow these solved questions, it will clear all doubts. The questions in various competitive exams can be confusing, and the Solutions provides an easy way for students to understand concepts by breaking them down into steps and explaining them enough to understand and be informed of the various questioning forms.