Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = 2^{n}$.

The geometric mean of $n$ positive terms $x_1, x_2, \dots, x_n$ is equal to:

There are two such pairs of non-zero real values of $a$ and $b$,i.e.,$(a_1, b_1)$ and $(a_2, b_2)$,for which $2a+b, a-b, a+3b$ are three consecutive terms of a $G.P.$. Then the value of $2(a_1b_2 + a_2b_1) + 9a_1a_2$ is-

The sum of the first two terms of a geometric progression is $12$. The sum of the third and fourth terms is $48$. The terms of the geometric progression alternate between positive and negative. What is the first term?

If $x$ is added to each of the numbers $3, 9, 21$ so that the resulting numbers are in $G.P.$,then the value of $x$ is:

If $1 + \cos \alpha + \cos^2 \alpha + \dots \infty = 2 - \sqrt{2}$,then $\alpha$ $(0 < \alpha < \pi)$ is