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

If the $(m + 1)^{th}$,$(n + 1)^{th}$,and $(r + 1)^{th}$ terms of an arithmetic progression are in geometric progression,and $m, n, r$ are in harmonic progression,then what is the ratio of the common difference to the first term of the arithmetic progression?

Two sequences $\{t_n\}$ and $\{s_n\}$ are defined by $t_n = \log \left( \frac{5^{n+1}}{3^{n-1}} \right)$ and $s_n = \left[ \log \left( \frac{5}{3} \right) \right]^n$. Then:

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

If three unequal numbers $p, q, r$ are in $H.P.$ and their squares are in $A.P.$,then the ratio $p:q:r$ is

Let $x_1, x_2, \ldots, x_{100}$ be in an arithmetic progression,with $x_1 = 2$ and their mean equal to $200$. If $y_i = i(x_i - i)$ for $1 \leq i \leq 100$,then the mean of $y_1, y_2, \ldots, y_{100}$ is