Sum of First N Odd Numbers

Sum of first n Odd Numbers is calculated by adding together integers that are not divisible by 2 from 1 to n, it can be easily calculated by using the formula, resulting in a total that is either an odd number or even number. Sum of first n Odd Numbers is often represented by the formula expressed as n² where n is a natural number. This formula can be used to calculate the sum of the first n odd numbers without adding them individually.

In this article, we will learn about the Sum of first n Odd Numbers Formula including the definition of Odd Numbers as well as some solved examples using the formula.

What are Odd Numbers?

Odd Numbers are integers that cannot be exactly divided by 2. In other words, when an odd number is divided by 2, it results in a fraction or a number with a remainder. This is in contrast to even numbers, which are divisible by 2 without any remainder.

Examples of Odd Numbers

Some of the common examples of Odd Numbers are:

• Small Odd Numbers: These are the odd numbers you encounter first when you start counting. Examples include 1, 3, 5, 7, and 9.
• Odd Numbers in the Double Digits: As you move into larger numbers, you still find odd numbers regularly. Examples include 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, etc.
• Large Odd Numbers: Odd numbers can be very large as well. Examples include 101, 333, 547, 2,019, 5,021, etc.
• Negative Odd Numbers: Odd numbers can also be negative. Examples include -1, -3, -5, -7, -9, -11, etc.

👁 Sum of Odd Numbers is N2

Read More about Odd Numbers.

How to Find the Sum of first n Odd Numbers?

In Sum of first n Odd numbers, the mathematical calculations are based on adding n consecutive Odd numbers together, or adding any of the odd numbers together. Even though, We know that these Odd numbers are not divisible by 2. There is a consequent that if we addend any two odd numbers together we will get Sum in even number. Let's see few examples on this terms.

• 3 + 5 = 8
• 5 + 7 = 12
• 5 + 1 = 6

Read More about Sum of N Terms of an AP.

Sum of n Odd Numbers Formula

The Sum of first n Odd Numbers can be expressed using a formula. If you want to find the sum of the first n odd numbers, the formula is:

S = n²

Where,

• S is the sum of the first n odd numbers, and
• n is the number of odd numbers you want to add.

Proof of Sum of first n Odd Numbers Formula

The odd numbers are 1, 3, 5, 7, 9, 11, . . . from the header above, as can be seen. If students examine closely, they can find an arithmetic progression sequence (AP) with common difference of 2 . The AP can be included into the formula in the following ways:

In first step, Lets see the simple formula of Sum of first n Odd Numbers:

(2n±1) It can represented as 2n+1 or 2n- 1

• In 2n+1 it can be begins with n= 0, 1, 3, 5, 7, 9 or so on.
• In 2n- 1 it can be begins with n= 1, 3, 5, 7, 9 or so on.
• For Ex- n= 0, 2n+1 becomes 1 or n= 1, 2n- 1 than also it becomes 1.
• It can be written as S_n = 1 + 3 + 5 + 7 + … + (2n−1).

In Second Step, Lets see the AP Formula:

S_n = n/2×(2a+(n−1)d)

• n is the number of terms,
• a is the first term
• d is the common difference

In Third Step, How we apply AP in Odd numbers:

• Here, a is the first term is 1
• And d is the common difference is 2.
• Even, the last term L is 2n- 1.

At last Substituting values of AP:

S_n = (n/2) × (1 + 2n – 1)

After Simplifying we get:

S_n= (n/2) × (2n) = n²

So, Sum of first n Odd Numbers in each terms is n².

Sum of first n Odd Numbers from 1 to 100

To find the Sum of first n Odd Numbers from 1 to 100, you can use the formula for the sum of an arithmetic series:
Let's calculate n First;

n= (a_n - a)/2 + 1

• a = 1 (the first odd number),
• d = 2 (the common difference between consecutive odd numbers), and
• a_n is the last term of the number.

n= (99-10/2 +1

⇒ n= 98/2 + 1

⇒ n= 49+1

⇒ n= 50

Therefore, n = 50

Let's use the formula for the sum of an arithmetic series,

​S_n = n/2​ × (a₁ ​+ a_n​)

• S_n is the sum of the series,
• n is the number of terms in the series = 50
• a is the First odd number = 1
• d is the common difference = 2

S_n = 50/2 ×(2×1+(50−1)×2)

⇒ S_n = 25×(2+98)

⇒ S_n = 25×100

⇒ S_n = 2500

The Sum of first n Odd Numbers from 1 to 100 is also 2500.

Sum of first n Odd Numbers NOT Starting from 1

Lets say we have to find Sum of first n Odd Numbers not starting from 1 then formula of Sum of first n Odd Numbers not starting from 1 will be given by Sum of first (n+1) Odd Numbers Till - 1

⇒ Sum of first (n+1) Odd Numbers Till - 1

⇒(n+1)² -1

⇒(n² + 1 + 2n) -1

⇒(n²+2n) = n(n+2)

Read More,

• Odd and Even Numbers
• Even Numbers
• Prime Numbers

Solved Examples on Sum of first n Odd Numbers

Example 1: Find the sum of the first 7 odd numbers.

Solution:

Given: n is the number of terms in the series = 7,

a is the first odd number = 1, and

d is the common difference = 2

Now, substitute the values into the formula: S_n = n/2×(2a+(n−1)d)

S_n = 7/2 × [2×1 + (7−1)×2]

⇒ S_n = 7/2 × [2+12]

⇒ S_n = 7/2 × 14

⇒ S_n = 49

Therefore, the sum of the first 7 odd numbers is 49.

Example 2: Find the Sum of first n Odd Numbers between 1 to 20.

Solution:

The odd numbers between 1 and 30 are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

Sum of first n Odd Numbers = 1 + 3+ 5 + 7 + 9 + 11 + 13+ 15 + 17 + 19 = 100

Hence, the Sum of first n Odd Numbers between 1 to 20 is 100.

Example 3: Seema has 5 Pencils. He bought 3 more Pencils. How many Pencils does Seema have?

Solution:

Seema has 5 pencils.

He bought 3 more pencils.

Total Pencil = 5 + 3 pencils

Thus, Total Pencil = 8 pencils

So, the total number of pencils = 8

Example 4: Find the Sum of first n Odd Numbers between 1 to 30.

Solution:

The odd numbers between 1 and 30 are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.

Sum of first n Odd Numbers = 1 + 3+ 5 + 7 + 9 + 11 + 13+ 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 = 225.

Hence, the Sum of first n Odd Numbers between 1 to 30 is 225.

Example 5: Add any two Consecutive Odd numbers, You will get even number. Justify this statement.

Solution:

Let's take two odd numbers = 3,5

Add both the numbers = 3 + 5

= 8

Hence, 8 is an even number because it is divisible by 2.

Therefore, the above statement is also justified, the addition of two consecutive numbers will give an even number.

Example 6: Find the sum of the first 5 odd numbers using the formula S_n= 1/2×n(2a+(n−1)d).

Solution:

So, here the sum in arithmetic series is given as S_n= 1/2×n(2a+(n−1)d).

Given: n is the number of terms in the series = 5,

a is the first odd number = 1, and

d is the common difference = 2

Now, substitute the values into the formula:

S₅ = 5/2 × [2×1 + (5−1)×2]

⇒ S₅ = 5/2 × [2+8]

⇒ S₅ = 5/2 × 10

⇒ S₅ = 25

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

Sum of first n Odd Numbers - Practice Questions

Q1: What is the sum of the first 10 Odd numbers?

Q2: Is 8 is an Odd number.

Q3: Derive this equation Sn= (n/2) × (1 + 2n – 1).

Q4: Sagar has 5 Pens. He bought 3 more Pens. How many Pens does Sagar have?