Question

If in an A.P. the sum of p terms is equal to sum of q terms, then prove that the sum of $p+q$ terms is zero.

Answer 1

We are given that $S_p=S_q$.
$\therefore \frac{P}{2}[2a+(p-1\left)d\right]=\frac{q}{2}[2a+(q-1\left)d\right]$
or $(2a-d)(p-q)+(p^2-q^2)d=0$,
cancel $p-q$ as $p\ne q$
or $2a-d+(p+q)d=0$
or $2a+(p+q-1)d=0$ .$\left(1\right)$
$\therefore S_{p+q}=\frac{p+q}{2}[2a+(p+q-1\left)d\right]$
$=\frac{p+q}{2}.0=0$, by $\left(1\right)$.