Let $a, b$ and $c$ be in $G.P.$ with common ratio $r,$ where $a \ne 0$ and $0 < r \le \frac{1}{2}.$ If $3a, 7b$ and $15c$ are the first three terms of an $A.P.,$ then the $4^{th}$ term of this $A.P.$ is

If $a_1 = a_2 = 2$ and $a_n = a_{n-1} - 1$ for $n > 2$,then $a_5$ is

If $x = \sum_{n=0}^{\infty} (-1)^{n} \tan^{2n} \theta$ and $y = \sum_{n=0}^{\infty} \cos^{2n} \theta$ for $0 < \theta < \frac{\pi}{4}$,then:

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:

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means between two distinct positive numbers $a$ and $b$. Then $G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2$ is equal to

Let the first term of a series be $T_1=6$ and its $r^{\text{th}}$ term $T_r=3T_{r-1}+6^r$ for $r=2, 3, \ldots, n$. If the sum of the first $n$ terms of this series is $\frac{1}{5}(n^2-12n+39)(4 \cdot 6^n - 5 \cdot 3^n + 1)$,then $n$ is equal to: