Question

In an A.P., of which a is the first term, if the sum of the first p terms is zero, show that the sum of the next q terms is $-\frac{a(p+q)q}{p-1}$.

Answer 1

$S_p=0$
$\therefore \frac{P}{2}[2a+(p-1\left)d\right]=0$
$\therefore d=\frac{-2a}{p-1}$ .$\left(1\right)$
Sum of next q terms $=$sum of an A.P. whose first term will be $T_{p+1}=a+pd$
$\therefore S=\frac{q}{2}\left[2\right(a+pd)+(q-1\left)d\right]$
$=\frac{q}{2}[2a+(p-1)d+(p+q\left)d\right]$
$=\frac{q}{2}[0-(p+q\left)\frac{2a}{p-1}\right]=-a\frac{(p+q)q}{p-1}$, by $\left(1\right)$.