In a geometric progression with positive terms,if each term is equal to the sum of the next two terms,then the common ratio of the progression is = .......

If $p, q, r$ are in one geometric progression and $a, b, c$ are in another geometric progression,then $cp, bq, ar$ are in

If five $G.M.s$ are inserted between $486$ and $2/3$,then the fourth $G.M.$ will be:

Let $a_{n}$ be the $n^{\text{th}}$ term of a $G$.$P$. of positive terms. If $\sum_{n=1}^{100} a_{2n+1} = 200$ and $\sum_{n=1}^{100} a_{2n} = 100$,then $\sum_{n=1}^{200} a_{n}$ is equal to:

$A$ 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 Geometric Progression $(GP)$ is $1$ and the sum of its third and fifth terms is $90$,find the common ratio.