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 $T_n=\lambda \left(2n-1\right)$
$\therefore T_m:T_n$
$=\left(2m-1\right):\left(2n-1\right)$