The sum of a $G.P.$ with common ratio $3$ is $364$, and last term is $243$, then the number of terms is

Find a $G.P.$ for which sum of the first two terms is $-4$ and the fifth term is $4$ times the third term.

If $y = x + {x^2} + {x^3} + .......\,\infty ,\,{\rm{then}}\,\,x = $

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

Let $P(x)=1+x+x^2+x^3+x^4+x^5$. What is the remainder when $P\left(x^{12}\right)$ is divided by $P(x)$ ?

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

$a^{q-r} b^{r-p} c^{p-q}=1$