Sum of squares of first n natural numbers means sum of the squares of the given series of natural numbers. The sum of squares of n natural numbers can be calculated using the formula [n(n+1)(2n+1)] /6. Let n be a natural number. Squaring the number is denoted by n2. Sum of Squares of First n Natural Numbers gives a generalized equation to get a sum of squares n Natural Numbers.

Natural numbers are those numbers used for counting and ordering. Natural numbers include positive integers. Hence, they are also known as non-negative integers. Natural numbers start from 1 and go up to infinity. 1, 2, 3, 4, 5, 6, … ∞

Maths Notes Free PDFs

 Sum of Squares of First n Natural Numbers

Sum of squares of n natural numbers means the sum of the squares of the given series of natural numbers. It could be finding the sum of squares of 2 numbers or 3 numbers or sum of squares of consecutive n numbers or n even numbers or n odd numbers.

The sum of squares of n natural numbers can be calculated using the formula [n(n+1)(2n+1)] / 6. Let n be a natural number. Squaring the number is denoted by n2.

If n consecutive natural numbers are 1, 2, 3, 4, …, n, then sum of squares of first n natural numbers is represented by \(1^2 + 2^2 + 3^2 + 4^2 + 5^2 + …….. + n^2\)

The sum of squares of n natural numbers can be calculated using the formula [n(n+1)(2n+1)] / 6. Let n be a natural number. Squaring the number is denoted by n2.

Formulas for finding the sum of squares of n natural numbers, the sum of squares of first n even numbers, and the sum of squares of first n odd numbers:

• Formula for Sum of squares of n natural numbers: [n(n+1)(2n+1)] / 6
• Formula for Sum of squares of first n even numbers: [2n(n + 1)(2n + 1)] / 3
• Formula for Sum of squares of first n odd numbers: [n(2n+1)(2n-1)] / 3

Sum of Squares Formulas and Proofs of First n Natural Numbers

Let us see the proof of Sum of Squares of First n Natural Numbers.

Sum of the squares of first n natural numbers = n(n+1)(2n+1)6.

Proof: Let us assume the required sum = S

Sum of Squares of First n Odd Natural Numbers

Now let’s see the Sum of the Squares of First n Odd Natural Numbers. Odd numbers are defined as numbers that are not exactly divisible by two. Or, to put it another way, an odd number is one that is not even or divisible by two. The numbers 1, 3, 5, 7, and 9 are odd numbers.

Squares of First 20 Natural Numbers

Sum of Squares in Geometry:

In a right-angled triangle, the longest side (the one opposite the right angle) is called the hypotenuse. According to the Pythagoras Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So,
Hypotenuse² = Base² + Height²

This idea helps us understand how lengths are related in right-angled triangles.

Sum of Squares of Two and Three Numbers

When dealing with small numbers, finding their squares and adding them is simple. But if you're working with bigger numbers or need to save time, some easy algebra formulas—called identities—can help.

For Two Numbers

If you have two numbers, say a and b, and you want to find a² + b², you can use:

a² + b² = (a + b)² − 2ab
This means:

1. Add a and b
    

2. Square the result
    

3. Subtract 2 × a × b

For Three Numbers

If you have three numbers a, b, and c, there's also a shortcut formula:
a² + b² + c² = (a + b + c)² − 2(ab + bc + ca)
It saves time and effort when adding squares of three numbers.

Important Notes on the Sum of Squares of Natural Numbers

When we talk about the sum of squares, we are adding the square of each number. For example, the square of 1 is 1, the square of 2 is 4, and so on.

1. Sum of Squares of First n Natural Numbers

If you want to add the squares of the first n natural numbers, the formula is:

1² + 2² + 3² + ... + n² = [n(n + 1)(2n + 1)] / 6

This formula helps you find the total quickly without having to square and add each number one by one.

2. Sum of Squares of First n Odd Numbers

Odd numbers look like 1, 3, 5, 7, etc. Their squares can also be added using a shortcut:

Sum = [n(2n + 1)(2n − 1)] / 3

3. Sum of Squares of First n Even Numbers

Even numbers are 2, 4, 6, 8, etc. For their squares, use this formula:

Sum = [2n(n + 1)(2n + 1)] / 3

Sum of Squares of n Natural Numbers Examples

Some solved examples on the sum of squares of natural numbers are given below:

Example 1: Find the Sum of Squares of First 8 Natural Numbers.

Solution: n = 8.

We know that, the sum of the squares of first n natural numbers = n(n+1)(2n+1)/6

Put n = 8 in this formula.

Therefore we get,

=204.

Example 2: What is the sum of square first 5 odd natural numbers?

Solution: n = 5

We know that, the sum of the squares of first 5 odd natural numbers = (n(2n+1)(2n−1)/3

Put n = 5 in this formula.

Therefore we get,

We hope you found this article regarding Sum of Squares of First n Natural Numbers was informative and helpful, and please do not hesitate to contact us for any doubts or queries regarding the same. You can also download the Testbook App, which is absolutely free and start preparing for any government competitive examination by taking the mock tests before the examination to boost your preparation. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: