Question

The sum of the first p terms of an A.P is q and the sum of the first q terms is p. The sum of the first (p + q) terms is

• A. 0
• B. (p + q)
• C. -(p + q)
• D. -2(p + q)

Answer 1

The correct option is C

-(p + q)

Let a, a + d, a + 2d, ….. be the A.P.

$S_p=q\Rightarrow \frac{p}{2}[2a+(p-1\left)d\right]=q$ . . . .(1)

$S_q=p\Rightarrow \frac{p}{2}[2a+(q-1\left)d\right]=p$ . . . .(2)

$\left(1\right)-\left(2\right)\Rightarrow \frac{(p-q)}{2}(2a-d)+\frac{d}{2}(p^2-q^2)=q-p$

$\Rightarrow \frac{2a-d}{2}+\frac{(p+q)d}{2}+=-cancellingp-q$

$or,2a+(p+q-1)d=-2$ .... (3)

$Now{S}_{p,q}=\frac{(p+q)}{2}[2a+(p+q-1\left)\right]=\frac{(p+q)}{2}\times (-2)=-(p+q)$