Question

If the sum of first $p$ terms of an A.P. is equal to the sum of first $q$ terms then show that the sum of its first $(p+q)$ terms is zero. $(p\ne q)$.

Answer 1

$S_p=S_q$

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

$\Rightarrow$ $p(2a+(p-1\left)d\right)=q(2a+(q-1\left)d\right)$

$\Rightarrow$ $2ap+p^{2}d-pd=2aq+q^{2}d-qd$

$\Rightarrow$ $2a(p-q)+(p+q)(p-q)d-d(p-q)=0$

$\Rightarrow$ $(p-q)[2a+(p+q)d-d]=0$

$\Rightarrow$ $2a+(p+q)d-d=0$

$\Rightarrow$ $2a+\left(\right(p+q)-1)d=0$

$\Rightarrow$ $S_{p+q}=0$