If $x,\;y,\;z$ are in $G.P.$ and ${a^x} = {b^y} = {c^z}$, then

Let $x _{1}, x _{2}, x _{3}, \ldots ., x _{20}$ be in geometric progression with $x_{1}=3$ and the common ration $\frac{1}{2}$. A new data is constructed replacing each $x_{i}$ by $\left(x_{i}-i\right)^{2}$. If $\bar{x}$ is the mean of new data, then the greatest integer less than or equal to $\bar{x}$ is $.....$

There are two such pairs of non-zero real valuesof $a$ 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-

Suppose the sides of a triangle form a geometric progression with common ratio $r$. Then, $r$ lies in the interval

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 

If $a,\;b,\;c$ are in $A.P.$, then ${10^{ax + 10}},\;{10^{bx + 10}},\;{10^{cx + 10}}$ will be in