Question

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

$S_p=q$ and $S_q=p$

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

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

$2a(p-q)+d\left\{\right(p^2-q^2)-(p-q\left)\right\}=2(q-p)$

$2a+d(p+d-1)=-2$

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

$=\frac{p+q}{2}(-2)=-(p+q)$.