Q69 / 180 · KCET 2024 · Mathematics
If $S_n$ stands for sum to $n$-terms of a GP with $a$ as the first term and $r$ as the common ratio, then $S_n: S_{2 n}$ is
- A$r^n+1$
- Correct answerB$\frac{1}{r^n+1}$
- C$r^n-1$
- D$\frac{1}{r^n-1}$
Solution
$S_n=\frac{a\left(1-r^n\right)}{1-r}$
$S_{2 n}=\frac{a\left(1-r^{2 n}\right)}{1-r}$
$\frac{S_n}{S_{2 n}}=\frac{a\left(1-r^n\right)}{1-r} \times \frac{1-r}{a\left(1-r^{2 n}\right)}=\frac{1-r^n}{\left(1+r^n\right)\left(1-r^n\right)}$
Hence, $S_n: S_{2 n}=\frac{1}{1+r^n}$
Hence, the correct answer is option: 'B'.