If $a, b, c$ are three consecutive terms of an $A.P.$ and $x, y, z$ are three consecutive terms of a $G.P.$,then the value of $x^{b-c} \cdot y^{c-a} \cdot z^{a-b}$ is:

Let $a = \min \{x^{2} + 2x + 3 : x \in R\}$ and $b = \lim_{\theta \rightarrow 0} \frac{1 - \cos \theta}{\theta^{2}}$. Then $\sum_{r=0}^{n} a^{r} b^{n-r}$ is

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:

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

Write the first five terms of the following sequence and obtain the corresponding series:
$a_{1} = a_{2} = 2, a_{n} = a_{n-1} - 1, n > 2$

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers in arithmetic progression with common difference $2$. Also,let $b_1, b_2, b_3, \ldots$ be a sequence of positive integers in geometric progression with common ratio $2$. If $a_1 = b_1 = c$,then the number of all possible values of $c$,for which the equality $2(a_1 + a_2 + \ldots + a_n) = b_1 + b_2 + \ldots + b_n$ holds for some positive integer $n$,is: