What is the solution to the equation $8^{(1 + |\cos x| + |\cos^2 x| + |\cos^3 x| + \dots)} = 4^3$ in the interval $(-\pi, \pi)$?

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

Three unequal positive numbers $a, b, c$ are such that $a, b, c$ are in $G.P.$ while $\log \left(\frac{5 c}{2 a}\right), \log \left(\frac{7 b}{5 c}\right), \log \left(\frac{2 a}{7 b}\right)$ are in $A.P.$ Then $a, b, c$ are the lengths of the sides of

Let $S$ be the sum of the first $9$ terms of the series: $(x+ka) + (x^2+(k+2)a) + (x^3+(k+4)a) + (x^4+(k+6)a) + \ldots$ where $a \neq 0$ and $x \neq 1$. If $S = \frac{x^{10}-x+45a(x-1)}{x-1}$,then $k$ is equal to:

If $\log_x y, \log_z x, \log_y z$ are in $G.P.$,$xyz = 64$,and $x^3, y^3, z^3$ are in $A.P.$,then