If $s$ is the sum of an infinite $G.P.$ and $a$ is the first term,then the common ratio $r$ is given by:

Insert three numbers between $1$ and $256$ so that the resulting sequence is a $G.P.$

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

$A$ manufacturer reckons that the value of a machine,which costs him $Rs. 15625$,will depreciate each year by $20\%$. Find the estimated value at the end of $5$ years.

If $\alpha, \beta$ are the roots of $x^2-3x+a=0$ and $\gamma, \delta$ are the roots of $x^2-12x+b=0$ and $\alpha, \beta, \gamma, \delta$ in that order form a geometric progression in increasing order with common ratio $r>1$,then $a+b=$

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