Sum of Squares of Even and Odd Natural Numbers
Formula for sum of square of even and odd number is covered below:
Sum of Squares of Even Natural Numbers
Formula for sum of square of even natural number is:
Sum = 2n(n+1)(2n+1)/3
This is calculated as:
- Sum = 22+ 42+ 62 +… + (2n)2
- Taking 22 or 4 commons from each term
- Sum = 4 × {12 + 22 + 32 +…+ (n)2}…(i)
- We know that sum of ‘n ‘Natural Number is given by:
- 12 + 22 + 32 +…+ (n)2 = n(n + 1)(2n + 1)/6…(ii)
- From eq. (i) and eq. (ii)
- Sum = 4n(n+1)(2n+1)/6
Sum = 2n(n+1)(2n+1)/3
Example: Find the sum of squares of the first 3 even numbers.
We know that,
Sum = 2n(n+1)(2n+1)/3
n = 3
Sum = 2×3(3+1)(2×3+1)/3
Sum = 56…(i)
Verification:
Sum = 22 + 42 + 62
⇒ Sum = 4 + 16 + 36
⇒ Sum = 56…(ii)
From eq(i) and eq(ii) Formula is Verified
Sum of Squares of Odd Natural Numbers
Formula for sum of square of odd natural number is:
Sum = n(2n+1)(2n-1)/3
This is calculated as:
- Sum = 12+ 32+ 52 +… + (2n – 1)2
- ⇒ Sum = (Sum of Squares of 2n numbers) – (Sum of Squares of First n Even Numbers)
- ⇒ Sum = 2n×(2n+1)×(2×2n + 1)/6 – 2n(n+1)(2n+1)/3
- ⇒ Sum = 2n(2n+1)/6 [4n+1 – 2(n+1)]
- ⇒ Sum = n(2n+1)/3 × (2n-1)
Sum = n(2n+1)(2n-1)/3
Example: Find the sum of squares of the first 3 odd numbers.
We know that,
Sum = n(2n+1)(2n-1)/3
n = 3
⇒ Sum = 3(2×3 + 1)(2×3 – 1)/3
⇒ Sum = 35…(i)
Verification:
Sum = 12 + 32 + 52
⇒ Sum = 1 + 9 + 25
⇒ Sum = 35…(ii)
From eq(i) and eq(ii) Formula is Verified
Sum of Squares of n Natural numbers
Sum of Squares of n Natural numbers: The sum of squares of n natural numbers is calculated using the formula [n(n+1)(2n+1)] / 6 where ‘n’ is a natural number. There are formulas for calculating the sum of squares of first n even numbers as well as first n odd numbers.
In this article, we have covered the sum of squares of the ‘n’ natural number formula, its sum of squares proof, related examples, and others in detail.
Sum of Squares of ‘n’ Natural Number Formula
Table of Content
- What is the Sum of Squares of ‘n’ Natural Numbers?
- Sum of Squares of n Natural Numbers Formula
- Sum of Squares of Natural Numbers Proof
- Sum of Squares of Even and Odd Natural Numbers
- Sum of Squares of Even Natural Numbers
- Sum of Squares of Odd Natural Numbers
- Sum of Squares of Two and Three Natural Numbers
- Sum of Squares in Geometry
- Sum of Squares of n Natural Numbers Solved Questions
- Practice Questions on Sum of Squares of n Natural Numbers
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