Question

If the sum of p terms of an AP is q and the sum of q term is p, then the sum of p + q terms will be:

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

Answer 1

The correct option is D -(p + q)

let first term be a and common difference be b
so $p^{th}$ term $=a+(p-1)d$
and sum of first p terms $=\frac{p}{2}(2a+(p-1\left)d\right)=q$
hence $(2a+(p-1\left)d\right)=\frac{2q}{p}....\left(1\right)$
and $q^{th}$ term $=a+(q-1)p$
and sum of first q terms $=\frac{q}{2}(2a+(q-1\left)d\right)=p$
hence $(2a+(q-1\left)d\right)=\frac{2p}{q}....\left(2\right)$
subtract (1) from (2)
$(p-q)d=2(\frac{q}{p}-\frac{p}{q})=\frac{2(q^2-p^2)}{q}$
so, $d=-\frac{2(p+q)}{pq}....\left(3\right)$
$(p+q{)}^{th}$ term $=(a+(p+q-1\left)d\right)$
and sum of its first $(p+q)$ term $=\frac{(p+q)}{2}(2a+(p+q-1\left)d\right)$
$=\frac{(p+q)}{2}(2a+(p-1)d+qd)=(\frac{2q}{p}+qd)\frac{(p+q)}{2}$... from(3)
$=(\frac{2q}{p}-2-\frac{2q}{p})\frac{p+q}{2}$
$=-(p+q)$