UPSC NDA Math Model Paper 6

Here we have to find the sum if the n term of an infinite series: 1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4 + ......
Here , the nth term of the given series is given by: an=n⋅(n+1)
So, the sum of the nth term of the given series is given by : Sn=‌

n

∑

k=1

ak=‌

n

∑

k=1

[k.(k+1)]
=

n

∑

k=1

k+

n

∑

k=1

k2
As we know that, ‌∑n=

n(n+1)

2

and ∑n2=

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

6

⇒Sn=

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

3