Let $a_{n}$ be the $n^{\text {th }}$ term of a G.P. of positive terms.

If $\sum\limits_{n=1}^{100} a_{2 n+1}=200$ and $\sum\limits_{n=1}^{100} a_{2 n}=100,$ then $\sum\limits_{n=1}^{200} a_{n}$ is equal to

- [JEE MAIN 2020]

- A
$225$

- B
$175$

- C
$300$

- D
$150$

Find the value of $n$ so that $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ may be the geometric mean between $a$ and $b .$

If $x,\,2x + 2,\,3x + 3,$are in $G.P.$, then the fourth term is

**In **a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals

- [AIEEE 2007]

If the sum of an infinite $G.P.$ and the sum of square of its terms is $3$, then the common ratio of the first series is

If the roots of the cubic equation $a{x^3} + b{x^2} + cx + d = 0$ are in $G.P.$, then