The sum of the first three terms of a $G.P.$ is $\frac{39}{10}$ and their product is $1$. Find the common ratio and the terms.

If the $p^{\text{th}}$,$q^{\text{th}}$,and $r^{\text{th}}$ terms of a $G.P.$ are $a$,$b$,and $c$ respectively,prove that $a^{q-r} b^{r-p} c^{p-q} = 1$.

$A$ person writes a letter to four of his friends. He asks each one of them to copy the letter and mail it to four different persons with the instruction that they continue the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter,find the total amount spent on postage when the $8^{\text{th}}$ set of letters is mailed.

What is the sum of the first $100$ terms of the series $0.9 + 0.09 + 0.009 + \dots$?

The third term of a geometric progression is equal to the square of the first term. If its second term is $8$,then its sixth term will be:

The sum of a few terms of a geometric series is $728$. If the common ratio is $3$ and the last term is $486$,then the first term of the series will be: