$A$ 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.

Which term of the sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$ is $\frac{1}{19683}$?

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

If $a, b, c$ are the $p^{th}, q^{th}$ and $r^{th}$ terms of a $G.P.$,then $\left( \frac{c}{b} \right)^p \left( \frac{b}{a} \right)^r \left( \frac{a}{c} \right)^q$ is equal to

In an increasing geometric series,the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25$. Then,the sum of the $4^{\text{th}}$,$6^{\text{th}}$,and $8^{\text{th}}$ terms is equal to

Find the $12^{\text{th}}$ term of a $G.P.$ whose $8^{\text{th}}$ term is $192$ and the common ratio is $2$.