Question

In an AP, if the pth term is $\frac{1}{q}$ and qth term is $\frac{1}{p}$. Then, the sum of first pq term is

• A. $(pq+1)$
• B. $\frac{1}{2}(pq+1)$
• C. $\frac{1}{2}(pq-1)$
• D. None of these

Answer 1

The correct option is B $\frac{1}{2}(pq+1)$

Given $a_p=\frac{1}{q}$

$a+(p-1)d=\frac{1}{q}$

$aq+(pq-q)d=\mathrm{1......}\left(1\right)$

Similarly, we get

$ap+(pq-p)d=\mathrm{1......}\left(2\right)$

From (1) and (2), we get

$aq+(pq-q)d=ap+(pq-p)d$

$aq-ap=d[pq-p-pq+q]$

$a(q-p)=d(q-p)$

$\therefore a=d$

Equation (1) becomes,

$dq+pqd-dq=1$

$d=\frac{1}{pq}$

Hence, $a=\frac{1}{pq}$

Consider,

$S_{pq}=\frac{pq}{2}[2a+(pq-1\left)d\right]$

$=\frac{pq}{2}[2\left(\frac{1}{pq}\right)+(pq-1\left)\left(\frac{1}{pq}\right)\right]$

$=\frac{1}{2}[2+pq-1]$

$=\frac{1}{2}[pq+1]$