Question

If the sum of first n terms of an A.P. be equal to the sum of its first m terms, $(m\ne n)$, then the sum of its first (m+n) terms will be

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

As given $\frac{n}{2}\{2a+(n-1\left)d\right\}=\frac{m}{2}\{a+(m-1\left)d\right\}$
$\Rightarrow 2a(m-n)+d(m^2-m-n^2+n)=0$
$\Rightarrow (m-n)\{2a+d(m+n-1\left)\right\}=0$
$\Rightarrow 2a+(m+n-1)d=0,(\therefore m\ne n)$
$\therefore S_{m+n}=\frac{m+n}{2}\{2a+(m+n-1\left)d\right\}=\frac{m+n}{2}\left\{0\right\}=0.$