If the first term of a $G.P.$ $a_1, a_2, a_3, \dots$ is unity such that $4a_2 + 5a_3$ is least,then the common ratio of the $G.P.$ is

If the first and the $n^{\text{th}}$ term of a $G.P.$ are $a$ and $b$ respectively,and if $P$ is the product of $n$ terms,prove that $P^{2} = (ab)^{n}$.

The sum of the $3^{rd}$ and the $4^{th}$ terms of a $G.P.$ is $60$ and the product of its first three terms is $1000$. If the first term of this $G.P.$ is positive,then its $7^{th}$ term is

If the sum of an infinite $GP$ $a, ar, ar^{2}, ar^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150$,then the sum of $ar^{2}, ar^{4}, ar^{6}, \ldots$ is:

If the product of three consecutive terms of a $G.P.$ is $216$ and the sum of their products taken two at a time is $156$,then the numbers are:

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a $G.P.$ are $a$,$b$,and $c$ respectively,then the value of $a(b - c)\log a + b(c - a)\log b + c(a - b)\log c$ is: