If all the terms of an $A.P.$ are squared,then the new series will be in

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 $\langle a_n \rangle$ be a sequence such that $a_0 = 0, a_1 = \frac{1}{2}$ and $2a_{n+2} = 5a_{n+1} - 3a_n$ for $n = 0, 1, 2, 3, \ldots$. Then $\sum_{k=1}^{100} a_k$ is equal to:

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:

If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that the $4^{\text{th}}$ $A.M.$ is equal to the $2^{\text{nd}}$ $G.M.$,then $m$ is equal to:

Let four distinct positive numbers $a_1, a_2, a_3, a_4$ be in a geometric progression. Let $b_1 = a_1$,$b_2 = b_1 + a_2$,$b_3 = b_2 + a_3$,and $b_4 = b_3 + a_4$.
Statement-$I$: The numbers $b_1, b_2, b_3, b_4$ are neither in an arithmetic progression nor in a geometric progression.
Statement-$II$: The numbers $b_1, b_2, b_3, b_4$ are in a harmonic progression.