Question

If the ratio of the sums of $m$ and $n$ terms of an $A.P$. is $m^2 : n^2$, then the ratio of its $m^{th}$ and $n^{th}$ terms is

• A. $\left(2m +1\right) : \left(2n +1 \right)$
• B. $\left(2m -1\right) : \left(2n - 1 \right)$
• C. $ m : n$
• D. $\left(m -1\right) : \left(n - 1 \right)$

Answer 1

Answer: (B)

$\frac{S_{m}}{S_{n}} = \frac{m^{2}}{n^{2}}$
$\Rightarrow \frac{S_{m}}{m^{2}} = \frac{S_{n}}{n^{2}} = \lambda$ (say).
If $T_{r}$ be the $r^{th}$ term, then $T_{m} = S_{m} -S_{m-1} $
$\Rightarrow T_{m} = \lambda\left(m^{2} -\left(m^{2}-1\right)^{2}\right) $
$= \lambda\left(2m - 1\right)$.
Similarly $T_{n} = \lambda\left(2n - 1\right)$
$\therefore T_{m} : T_{n}$
$ = \left(2m -1 \right) : \left(2n - 1\right)$