Question

If $S_n$ stands for the sum to $n$-terms of a G.P. with $a$ as the first term and $r$ as the common ratio, then $\frac{S_1}{S_2}$ is:

• A. $r^n + 1$
• B. $1/r^n + 1$
• C. $r^n - 1$
• D. $\frac{1}{r^n - 1}$

Answer 1

The Correct Option is B

1. Understand the problem:

We are given a geometric progression (G.P.) with first term 'a' and common ratio 'r'. We need to find the ratio of the sum of the first n terms (\( S_n \)) to the sum of the first 2n terms (\( S_{2n} \)).

2. Recall the formula for sum of a G.P.:

The sum of the first n terms of a G.P. is given by:

\[ S_n = a \frac{1 - r^n}{1 - r} \quad \text{(for \( r \neq 1 \))} \]

Similarly, the sum of the first 2n terms is:

\[ S_{2n} = a \frac{1 - r^{2n}}{1 - r} \]

3. Compute the ratio \( \frac{S_n}{S_{2n}} \):

Divide \( S_n \) by \( S_{2n} \):

\[ \frac{S_n}{S_{2n}} = \frac{a \frac{1 - r^n}{1 - r}}{a \frac{1 - r^{2n}}{1 - r}} = \frac{1 - r^n}{1 - r^{2n}} \]

4. Simplify the expression:

Notice that \( 1 - r^{2n} \) can be factored as \( (1 - r^n)(1 + r^n) \). Thus:

\[ \frac{1 - r^n}{1 - r^{2n}} = \frac{1 - r^n}{(1 - r^n)(1 + r^n)} = \frac{1}{1 + r^n} \]

5. Match the result to the options:

The simplified form \( \frac{1}{1 + r^n} \) corresponds to option (B) \( \frac{1}{r^n + 1} \).

Correct Answer: (B) \( \frac{1}{r^n + 1} \)

Answer 2

If we use the formula for the sum of a geometric series with \( |r| < 1 \), then:

\[ S_n = \frac{a(1 - r^n)}{1 - r} \] \[ S_{2n} = \frac{a(1 - r^{2n})}{1 - r} \] \[ \frac{S_n}{S_{2n}} = \frac{1 - r^n}{1 - r^{2n}} = \frac{1 - r^n}{(1 - r^n)(1 + r^n)} = \frac{1}{1 + r^n} \]

If \( |r| > 1 \), then:

\[ S_n = \frac{a(r^n - 1)}{r - 1} \] \[ S_{2n} = \frac{a(r^{2n} - 1)}{r - 1} \] \[ \frac{S_n}{S_{2n}} = \frac{r^n - 1}{r^{2n} - 1} = \frac{r^n - 1}{(r^n - 1)(r^n + 1)} = \frac{1}{r^n + 1} \]