Find the sum of the products of the corresponding terms of the sequences $2, 4, 8, 16, 32$ and $128, 32, 8, 2, \frac{1}{2}$.

What is the product of $n$ geometric means between $a$ and $b$?

Find the sum to the indicated number of terms in the geometric progression: $x^{3}, x^{5}, x^{7}, \ldots$ for $n$ terms (where $x \neq \pm 1$).

What is the geometric mean of $3, 3^2, 3^3, \dots, 3^n$?

$A$ particle starts at the origin and moves $1$ unit horizontally to the right and reaches $P_{1}$,then it moves $\frac{1}{2}$ unit vertically up and reaches $P_{2}$,then it moves $\frac{1}{4}$ unit horizontally to the right and reaches $P_{3}$,then it moves $\frac{1}{8}$ unit vertically down and reaches $P_{4}$,then it moves $\frac{1}{16}$ unit horizontally to the right and reaches $P_{5}$ and so on. Let $P_{n} = (x_{n}, y_{n})$ and $\lim_{n \rightarrow \infty} x_{n} = \alpha$ and $\lim_{n \rightarrow \infty} y_{n} = \beta$. Then,$(\alpha, \beta)$ is

If the first term of a $G.P.$ is $5$ and the common ratio is $-5$,then which term is $3125$ (in $^{th}$)?