Sum of Squares of First n Natural, Even & Odd Numbers – Formulas & Proof
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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, … ∞
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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
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 n Natural Numbers Formula
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
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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
Number n
Square n2
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
11
121
12
144
13
169
14
196
15
225
16
256
17
289
18
324
19
361
20
400
|
Number n |
Square n2 |
|
1 |
1 |
|
2 |
4 |
|
3 |
9 |
|
4 |
16 |
|
5 |
25 |
|
6 |
36 |
|
7 |
49 |
|
8 |
64 |
|
9 |
81 |
|
10 |
100 |
|
11 |
121 |
|
12 |
144 |
|
13 |
169 |
|
14 |
196 |
|
15 |
225 |
|
16 |
256 |
|
17 |
289 |
|
18 |
324 |
|
19 |
361 |
|
20 |
400 |
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:
-
Add a and b
-
Square the result
-
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,
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FAQs For Sum of Squares of First N Natural Numbers
What is 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.
What is sum of squares of n natural numbers formula?
If n consecutive natural numbers are 1, 2, 3, 4, …, n, then the sum of squared 'n' consecutive natural numbers is represented by
What is the sum of squares of first n even natural numbers?
Sum of squares of first n even natural numbers can be written as
What is the sum of squares of first n odd natural numbers?
Sum of squares of first n odd natural numbers can be written as
What is the sum of squares of first 10 natural numbers?
n = 10.We know that the sum of the squares of first 10 even natural numbers =
Why does the formula for the sum of squares of natural numbers work?
The formula works because it is derived from the properties of arithmetic series and geometric progressions. It uses a combination of algebraic techniques to sum up squared terms efficiently, avoiding the need to add them one by one.
How is the sum of squares related to other mathematical concepts?
The sum of squares is related to series, sequences, and polynomials. It helps in solving problems in number theory, combinatorics, and statistics, especially when dealing with variance and standard deviation.