UPSC NDA Math Model Paper 9

Let's say that the first term of the AP is a and the common difference is d.
The sum of the first p terms (p ≠ 0) will be:
Sp=

p

2

[2a+(p−1)d]
According to the question:
Sp=0
⇒

p

2

[2a+(p−1)d]=0
⇒2a+(p−1)d=0 ...(1)
⇒d=−

2a

p−1

...(2)
The sum of the next q terms will be:
Sp+q−Sp
=Sp+q−0
=

p+q

2

[2a+(p+q−1)d]
Using the equations (1) and (2), we get:
=

p+q

2

[0+q(

−2a

p−1

)]
=−

aq(p+q)

p−1