Question

If the ratio of the sum of m and $n$ terms of an AP is $m^2:n^2,$ then the ratio of its mth and nth terms is:

• A. $m-1:n-1$
• B. $2m+1:2n+1$
• C. $2m-1:2n-1$
• D. none of these

Answer 1

Answer: (C)

Let a and d be the first term and common difference respectively. Given that
$\frac{S_m}{S_n}=\frac{m^2}{n^2}$
$\Rightarrow$ $\frac{m/2[2a+(m-1)d]}{n/2[2a+(n-1)d]}=\frac{m^2}{n^2}$
$\Rightarrow$ $\frac{2a+(m-1)d}{2a+(n-1)d}=\frac{m}{n}$
Replace $m$ by $2m-1$ and $n$ by $2n-1$
$\Rightarrow$ $\frac{2a+(2m-2)d}{2a+(2n-2)d}=\frac{2m-1}{2n-1}$
$\frac{a+(m-1)d}{a+(n-1)d}=\frac{2m-1}{2n-1}$
$\Rightarrow$ Ration of mth and nth terms
$=(2m-1):(2n-1)$