Question

If the ratio of $m^{th}$ and $n^{th}$ terms of an A.P. is $2m-1:2n-1$, then the ratio of sum of $m^{th}$ and $n^{th}$ terms of the series is

• A. $m:n$
• B. $m^2:n^2$
• C. $\sqrt{m}:\sqrt{n}$
• D. $m^2-1:n^2-1$

Answer 1

The correct option is B $m^2:n^2$

$\frac{S_{2m-1}}{S_{2n-1}}=\frac{(2m-1)[2a+(2m-2\left)d\right]}{(2n-1)[2a+(2n-2\left)d\right]}\frac{S_{2m-1}}{S_{2n-1}}=\frac{(2m-1)2(a+(m-1\left)d\right)}{(2n-1)2(a+(n-1\left)d\right)}\frac{S_{2m-1}}{S_{2n-1}}=\frac{(2m-1{)}^2}{(2n-1{)}^2}\Rightarrow \frac{S_m}{S_n}=\frac{m^2}{n^2}$