Q.8. A geometric sequence has first term $a$ and common ratio $r$, where $a$ and $r$ are positive integers and $r > 1$.

The sum of the first $n$ terms of this sequence is denoted by $S_n$.

It is given that the terms of the sequence satisfy:

$$S_{30} - S_{20} = k S_{10}$$

for some positive integer $k$. What is the smallest possible value of $k$?

Answer Choices

• $2^{10}$
• $2^{20}$
• $2^{30}$
• $\frac{2^{10}}{2^{10} - 1}$
• $2^{10}(2^{10} - 1)$

Solution

The formula for the sum of the first $n$ terms of a geometric sequence is:

$$S_n = a \frac{r^n - 1}{r - 1}$$

Substituting this into the given equation $S_{30} - S_{20} = k S_{10}$ and simplifying, we find:

$$r^{20} = k$$

Thus, the smallest possible value of $k$ is $r^{20}$. Since the smallest value of $r$ is 2, we have $k = 2^{20}$.

Therefore, the smallest possible value of $k$ is 2^{20}.