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. 0.0
• B. p - q
• C. p + q
• D. -(p + q)

Answer 1

The correct option is D -(p + q)

let first term be aand common difference be d
$so{p}^{th}term=a+(p-1)d$
$andsumoffirstpterms=\frac{p}{2}(2a+(p-1\left)d\right)=q$
$hence(2a+(p-1\left)d\right)=\frac{2q}{p}....\left(1\right)and{q}^{th}term=a+(q-1)d$
$andsumoffirstqterms=\frac{q}{2}(2a+(q-1\left)d\right)=phence(2a+(q-1\left)d\right)=\frac{2p}{q}....\left(2\right)subtract\left(1\right)from\left(2\right)(p-q)d=2(\frac{q}{p}-\frac{p}{q})=\frac{2(q^2-p^2)}{q}so,d=-\frac{2(p+q)}{pq}....\left(3\right)(p+q{)}^{th}term=(a+(p+q-1)d\left)andsumofitsfirst\right(p+q)term=\frac{(p+q)}{2}(2a+(p+q-1)d)=\frac{(p+q)}{2}(2a+(p-1)d+qd)=(\frac{2q}{p}+qd)\frac{(p+q)}{2}...from(1)=(\frac{2q}{p}-2-\frac{2q}{p})\frac{p+q}{2}...from(3)=-(p+q)$