Let $a, b, c > 1$. If $a^3, b^3, c^3$ are in $A.P.$ and $\log_a b, \log_c a, \log_b c$ are in $G.P.$,and the sum of the first $20$ terms of an $A.P.$ with first term $\frac{a+4b+c}{3}$ and common difference $\frac{a-8b+c}{10}$ is $-444$,then $abc$ is equal to:

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. The greatest value of $n$ for which $a_n$ is the greatest is:

Let $n \geq 4$ be a positive integer and let $l_1, l_2, \ldots, l_n$ be the lengths of the sides of an arbitrary $n$-sided non-degenerate polygon $P$. Suppose $\frac{l_1}{l_2} + \frac{l_2}{l_3} + \ldots + \frac{l_{n-1}}{l_n} + \frac{l_n}{l_1} = n$. Consider the following statements:
$I$. The lengths of the sides of $P$ are equal.
$II$. The angles of $P$ are equal.
$III$. $P$ is a regular polygon if it is cyclic.

Suppose the sequence $a_1, a_2, a_3, \ldots$ is an arithmetic progression of distinct numbers such that the sequence $a_1, a_2, a_4, a_8, \ldots$ is a geometric progression. The common ratio of the geometric progression is

Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled,then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of the $G.P.$ is $3r^{2}$,then $r^{2}-d$ is equal to:

The sum of the first ten terms of an $A$.$P$. is $160$ and the sum of the first two terms of a $G$.$P$. is $8$. If the first term of the $A$.$P$. is equal to the common ratio of the $G$.$P$. and the first term of the $G$.$P$. is equal to the common difference of the $A$.$P$.,then the sum of all possible values of the first term of the $G$.$P$. is: