Question

Suppose $a_1,a_2,a_3,...,a_n$ are in A.P and $S_k$ denotes the sum of first $k$ terms, If $\frac{S_n}{S_m}=\frac{n^3}{m^3}$, then the ratio of $(m+1{)}^{\text{th}}$ term to $(n+1{)}^{\text{th}}$ term is

• A. $\frac{(2m-1{)}^2}{(2n-1{)}^2}$
• B. $\frac{(2m+1{)}^2}{(2n+1{)}^2}$
• C. $\frac{(2m+1{)}^3}{(2n+1{)}^3}$
• D. $\frac{2m+1}{2n+1}$

Answer 1

The correct option is B $\frac{(2m+1{)}^2}{(2n+1{)}^2}$

Given,
$\frac{S_n}{S_m}=\frac{n^3}{m^3}$
$\Rightarrow \frac{T_m}{T_n}=\frac{(2m-1{)}^2}{(2n-1{)}^2}$
$\Rightarrow \frac{T_{m+1}}{T_{n+1}}=\frac{\left(2\right(m+1)-1{)}^2}{\left(2\right(n+1)-1{)}^2}$
$\Rightarrow \frac{T_{m+1}}{T_{n+1}}=\frac{(2m+1{)}^2}{(2n+1{)}^2}$