Sum of Squares of n Natural Numbers Solved Questions
Question 1: Find the sum of the squares of 9 and 11.
Solution:
Using formula a2 + b2 = (a +b)2 β 2ab,
= 92 + 112
= (9 + 11)2 β 2 Γ 9 Γ 11
= 202 β 198
= 400- 198 = 202β¦(i)
Verification
92 + 112 = 81 + 121 = 202β¦(ii)
From eq(i) and (ii) verified.
Question 2: Find 3 consecutive natural numbers if the sum of their squares is 50?
Solution:
Let the three number be, n, n+1 and n+2.
Given,
n2 + (n+1)2 +(n+2)2 = 50
β n2 + n2 + 2n + 1+ n2 + 4n + 4 = 50
β 3n2 + 6n + 5 = 50
β 3n2 + 6n = 45
Dividing by 3,
n2 + 2n = 15
β n2 + 2n β 15 = 0
β n2 + 5n β 3n β 15 = 0
β (n + 5)(n β 3) = 0
According to question, βnβ cannot be negative.
Thus, n β 3 = 0
β n = 3
Hence, required number are,
n, n+1 and n+2 = 3, 4, and 5
Question 3: Find the sum of the squares of the first 14 odd numbers.
Solution:
Formula for sum of the squares of n odd numbers = [n(2n+1)(2n-1)]/3
Here n = 14
[n(2n+1)(2n-1)]/3
= 14(29)(27)/3
= 3654
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 βnβ 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 βnβ natural number formula, its sum of squares proof, related examples, and others in detail.
Table of Content
- What is the Sum of Squares of βnβ 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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