A certain sequence is defined by the following rule: $$S_n=k(S_{n-1}),$$ where $k$ is a constant. If $$S_1=64\ \ \ \ and\ \ \ \ \ S_{25}=192,$$ what is the value of
$$S_9?$$

$$A.\ \ \ \sqrt{2}$$ $$B.\ \ \ \sqrt{3}$$ $$C.\ \ \ 64\sqrt{3}$$ $$D.\ \ \ 64\sqrt[3]{3}$$ $$E.\ \ \ 64\sqrt[24]{3}$$ Experts, how can I find the correct answer here? I need some help to solve this PS question.

Hi VJesus12,
Lets take a look at your question.

The sequence is defined by the rule:
$$S_n=k(S_{n-1})$$
Then
$$S_2=k(S_1)$$
$$S_3=k(S_2)=k(k(S_1))=k^2(S_1)$$
$$S_4=k(S_3)=k(k^2(S_1))=k^3(S_1)$$
$$S_5=k(S_4)=k(k^3(S_1))=k^4(S_1)$$
$$. . . . .$$
$$. . . . .$$
$$. . . . .$$
$$S_{25}=k(S_{24})=k(k^{23}(S_1))=k^{24}(S_1)$$
Therefore, we got a formula for S25,
$$S_{25}=k^{24}(S_1)$$
Plugin the known values:
$$192=k^{24}(64)$$
$$k^{24}=\frac{192}{64}=3$$
$$k^{\frac{24}{3}}=3^{\frac{1}{3}}$$
$$k^{8}=3^{\frac{1}{3}}$$
We are asked to find S9:
$$S_9=k^{8}(S_1)$$
$$S_9=3^{\frac{1}{3}}(64)$$
$$S_9=64\sqrt[3]{3}$$
Therefore, Option D is correct.

Hope it helps.
I am available if you'd like any follow up.