UPSC NDA Math Model Paper 4

We know that the sum of the squares of the first n natural numbers =

n(n+1)(2n+1)

6

.
∴ Mean of the squares of the first n numbers =

n(n+1)(2n+1) 6

n

=

(n+1)(2n+1)

6

According to the question: Mean = 11.
⇒

(n+1)(2n+1)

6

=11
⇒2n2+n+2n+1=66
⇒2n2+3n−65=0
⇒2n2+13n−10n−65=0
⇒n(2n+13)−5(2n+13)=0
⇒(2n+13)(n−5)=0
⇒2n+13=0 OR n−5=0
⇒n=−

13

2

ORn=5
Since n is a count, n=5.
Check: Sum of the squares of the first 5 natural numbers = 12+22+32+42+52 =1+4+9+16+25=55
Mean =55÷5=11.