Question

The sum of the first $n$ terms of an A.P. is half of the sum of the next $n$ terms. In the usual notation, the value $\frac{S_{3n}}{S_n}$ is

• A. $10$
• B. $8$
• C. $6$
• D. $4$

Answer 1

The correct option is C $6$

$\frac{S_{3n}}{S_n}$

$=\frac{\frac{3n}{2}(2a+(3n-1\left)d\right)}{\frac{n}{2}(2a+(n-1\left)d\right)}$

$=\frac{3\left(\right(n+1)d+(3n-1\left)d\right)}{(n+1)d+(n-1)d}$

$=\frac{3d(n+1+3n-1)}{d(n+1+n-1)}$

$=\frac{3\left(4n\right)}{2n}$

$=6$