The sum of odd numbers from 1 to infinity can be found easily, using Arithmetic Progression. As we know, the odd numbers are the numbers which are not divisible by 2. They are 1,3,5,7,9,11,13,15,17,19 and so on. Now, we need to find the sum of these numbers.

Let the sum of first n odd numbers be S_n

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) = n²

Sum of 1 to 10 odd numbers

Below is the table for the sum of odd numbers.

Also, read:

• Sum Of N Terms
• Sum Of Squares

Proof of Sum of Odd Numbers

To prove: Sum of ‘n’ consecutive odd numbers = n²

Step 1:

We need to understand the pattern of odd numbers sequence to prove their sum. The total of any set of sequential odd numbers beginning with 1 is always equal to the square of the number of digits, added together. If 1,3,5,7,9,11,…, (2n-1) are the odd numbers, then;

• Sum of first odd number = 1
• Sum of first two odd numbers = 1 + 3 = 4 (4 = 2 x 2).
• Sum of first three odd numbers = 1 + 3 + 5 = 9 (9 = 3 x 3).
• Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16 (16 = 4 x 4).

Step 2:

The number of digits added collectively is always equal to the square root of the total number.

• Sum of first odd number = 1.

The square root of 1, √1 = 1, so, only one digit was added.

• Sum of consecutive two odd numbers = 1 + 3 = 4.

The square root of 4, √4 = 2, so, two digits were added.

• Sum of first three consecutive odd numbers = 1 + 3 + 5 = 9.

The square root of 9, √9 = 3, so, three digits were added.

• Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16.

The square root of 16, √16 = 4, so, four digits were added.

Step 3:

Hence, from the above estimation, we can prove the formula to find the sum of the first n odd numbers is n x n or n².

• For example, if we put n = 21, then we have 21 x 21 = 441, which is equal to the sum of the first 21 odd numbers.

Note: If we don’t know the number of odd numbers present in a series, then the formula to determine the sum between 1 and n is (1/2(n + 1))².

Solved Examples

Question 1: What is the sum of odd numbers from 1 to 50?

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

Thus, n = 25

By the formula of sum of odd numbers we know;

S_n = n²

S_n = 25² = 625

Question 2: What is the sum of odd numbers from 1 to 99?

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

Thus, n = 50

By the formula of sum of odd numbers we know;

S_n = 50²

S_n = 50² = 2500

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