If three numbers are in $G.P.$,then their logarithms will be in

If $2^{10} + 2^{9} \cdot 3^{1} + 2^{8} \cdot 3^{2} + \ldots + 2^{1} \cdot 3^{9} + 3^{10} = S - 2^{11}$,then $S$ is equal to

The sum of an infinite geometric series is $20$,and the sum of the squares of its terms is $100$. Find the common ratio of the geometric series.

$A$ 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 the first term of an infinite geometric series is $x$ and its sum is $5$,then which of the following is true?

Find the fractional value of $0.125125125 \dots$