Question

If the sum of p terms of an A. P. is q and the sum of q terms is p, then the sum of p + q terms is ______.

• A. 0
• B. p-q
• C. p+1
• D. - (p + q)

Answer 1

The correct option is D - (p + q)

$S_p=\frac{p}{2}\times [2a+(p-1)d]=q........\left(i\right)S_q=\frac{q}{2}\times [2a+(q-1)d]=p........\left(ii\right)S_{p+q}=\frac{p+q}{2}\times [2a+(p+q-1)d]$

From equation (i) and (ii), we get,

$\frac{p}{2}\times [2a+(p-1)d]=q=>2a+(p-1)d=\frac{2q}{p}........\left(iii\right)and\frac{q}{2}\times [2a+(q-1)d]=p=>2a+(q-1)d=\frac{2p}{q}........\left(iv\right)$

Subtracting (iii) and (iv), we get,

$\frac{2p}{q}-\frac{2q}{p}=(q-p)d=>\frac{2(p+q)(p-q)}{pq}=(q-p)d=>d=\frac{-2(p+q)}{pq}........\left(v\right)$

Now, $S_{p+q}=\frac{p+q}{2}\times [2a+(p+q-1)d]=\frac{p+q}{2}\times [2a+(p-1)d+qd]=\frac{p+q}{2}\times [\frac{2q}{p}+q\left(\frac{-2(q+p)}{qp}\right)]=\frac{p+q}{2}\times \left(\frac{2q-2q-2p}{p}\right)=-(p+q)$