Question

Let $S_n$ denotes the sum of $n$ terms of an AP. If $S_{2n}=3S_n$, then the ratio $\frac{S_{3n}}{S_n}=$

• A. $4$
• B. $6$
• C. $8$
• D. $10$

Answer 1

The correct option is B

$6$

Determine the value of $\frac{S_{3n}}{S_n}$

We know that sum of $n$terms of an AP is $S_n=\frac{n}{2}\left[2a+(n-1)d\right]$, where $a$ be the first term and $d$ be the common difference of an AP.

Step 1: Determine the value of $a$

$$\begin{gathered} \Rightarrow S_{2n}=3S_n\left[Given\right] \\ \Rightarrow \frac{2n}{2}\left[2a+(2n-1)d\right]=\frac{3n}{2}\left[2a+(n-1)d\right] \\ \Rightarrow 4a+4nd-2d=6a+3nd-3d \\ \Rightarrow nd+d=2a \\ \Rightarrow 2a=nd+d.............\left(1\right) \end{gathered}$$

Step 2: Determine the value of $\frac{S_{3n}}{S_n}$

$$\begin{gathered} \frac{S_{3n}}{S_n}=\frac{{\displaystyle \frac{3n}{2}}\left[2a+(3n-1)d\right]}{{\displaystyle \frac{n}{2}}\left[2a+(n-1)d\right]} \\ =\frac{6a+9nd-3d}{2a+nd-d} \\ =\frac{3nd+3d+9nd-3d}{nd+d+nd-d}\left[\mathrm{From}\left(1\right)\right] \\ =\frac{12nd}{2nd} \\ =6 \end{gathered}$$

Therefore, the correct option is B