The sum of the series $5.05 + 1.212 + 0.29088 + \dots \infty$ is

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-

If the $4^{th}, 7^{th}$ and $10^{th}$ terms of a $G.P.$ are $a, b, c$ respectively,then the relation between $a, b, c$ is

Evaluate $\sum\limits_{k = 1}^{11} {\left( {2 + {3^k}} \right)} $

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 = .......

The first term of an infinite geometric progression is $x$ and its sum is $5$. Then: