Question

If in an A.P. the sum of the first $p$ terms and first $q$ terms are equal. Then prove that the sum of its first $p+q$ terms is $0$.

Answer 1

Let $a$ be the first term and $d$ be the common difference of the A.P.

Then according to the problem,

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

or, $2a(p-q)+(p+q-1)d=0$.....(1). [ Since $p$ and $q$ are distinct]

Now the sum of $p+q$ terms of the A.p. is

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

$=0$. [ Using (1)]