Question

If the sum of first 'p' terms of an A.P is equal to the sum of the first 'q' terms , then find the sum of the first (p + q ) terms .

Answer 1

Given $S_p=S_q$
$\Rightarrow \frac{p}{2}[2a+(p-1\left)d\right]=\frac{q}{2}[2a+(q-1\left)d\right]$
$\Rightarrow p[2a+(p-1\left)d\right]=q[2a+(q-1\left)d\right]$
$\Rightarrow 2ap+(p-1)pd=2aq+(q-1)qd$
$\Rightarrow 2a(p-q)+d(p^2-p-q^2+p)=0$
Now , sun of the first (p + q) terms is
$=\frac{(p+q)}{2}\left[0\right]\left(using\right(1\left)\right)$
= 0