If the product of three consecutive terms of $G.P.$ is $216$  and the sum of product of pair-wise is $156$, then the numbers will be

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 $…..$

If ${s_n} = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + …….. + \frac{1}{{{2^{n – 1}}}}$ , then the least integral value of $n$ such that $2 – {s_n} < \frac{1}{{100}}$ is

$\alpha ,\;\beta $ are the roots of the equation ${x^2} – 3x + a = 0$ and $\gamma ,\;\delta $ are the roots of the equation ${x^2} – 12x + b = 0$. If $\alpha ,\;\beta ,\;\gamma ,\;\delta $ form an increasing $G.P.$, then $(a,\;b) = $

Let $a_1, a_2, a_3, \ldots .$. be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, \ldots .$. be a sequence of positive integers in geometric progression with common ratio $2$ . If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality

$2\left(a_1+a_2+\ldots .+a_n\right)=b_1+b_2+\ldots . .+b_n$

holds for some positive integer $n$, is. . . . . . .