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. p - q
• B. p + q
• C. -(p + q)

Answer 1

The correct option is

B

-(p + q)

Let first term be aand common difference be b

So$p^{th}term=a+(p-1)d$

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

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

And$q^{th}term=a+(q-1)p$

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

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

Subtract$\left(1\right)from\left(2\right)$

$(p-q)d=2(\frac{q}{p}-\frac{p}{q})=\frac{2(q^2-p^2)}{q}$

so, $d=-\frac{2(p+q)}{pq}....\left(3\right)$

$(p+q{)}^{th}term=(a+(p+q-1)d)$

And$sumofitsfirst(p+q)term=\frac{(p+q)}{2}(2a+(p+q-1\left)d\right)$

$=\frac{(p+q)}{2}(2a+(p-1)d+qd)=(\frac{2q}{p}+qd)\frac{(p+q)}{2}...from\left(3\right)$

$=(\frac{2q}{p}-2-\frac{2q}{p})\frac{p+q}{2}$
$=-(p+q)$