UPSC NDA Math Model Paper 6

Here, we have to find the average of the squares of the numbers 0, 1, 2, ......, n
First let's find

n+1

∑

i=1

xi2=02+12+22+.......+n2
As we know that, 12+22+32+.......+n2 =∑n2=

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

6

⇒

n+1

∑

i=1

xi2=

n×(n+1)×(2n+1)

6

As we know that, the average of n observations x1,x2,.......,xn is given by:

‌ n ∑ i=1 xi

n

⇒

n+1 ∑ i=1 xi2

n+1

=

[n×(n+1)×(2n+1)6]

n+1

=

n(2n+1)

6

Hence, option B is the correct answer.