The product of three geometric means between $4$ and $\frac{1}{4}$ will be

Let $a, b, c, d$ and $p$ be any non-zero distinct real numbers such that $(a^{2}+b^{2}+c^{2}) p^{2} - 2(ab+bc+cd) p + (b^{2}+c^{2}+d^{2}) = 0$. Then:

Let $a, ar, ar^2, \ldots$ be an infinite $G.P.$ If $\sum_{n=0}^{\infty} ar^n = 57$ and $\sum_{n=0}^{\infty} a^3 r^{3n} = 9747$,then $a + 18r$ is equal to:

The sum of an infinite geometric series is $20$,and the sum of the squares of its terms is $100$. Find the common ratio of the geometric series.

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$,then $a_6$ is equal to

The greatest positive integer $k,$ for which $49^k+1$ is a factor of the sum $49^{125}+49^{124}+\ldots+49^{2}+49+1$ is