The roots of the equation $x^3-14x^2+56x-64=0$ are in

If $a, b, c, d$ and $p$ are different real numbers such that $(a^{2}+b^{2}+c^{2}) p^{2}-2(ab+bc+cd) p+(b^{2}+c^{2}+d^{2}) \leq 0$,then show that $a, b, c$ and $d$ are in $G.P.$

If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1 = \frac{1}{8}$ and $a_2 \neq a_1$ is the arithmetic mean of the next two terms and $S_n = a_1 + a_2 + \ldots + a_n$,then $S_{20} - S_{18}$ is equal to

Given a $G.P.$ with $a=729$ and $7^{\text{th}}$ term $64,$ determine $S_{7}$.

If $x, 2x + 2$,and $3x + 3$ are in a geometric progression,what is the fourth term?

If $x > 1, y > 1, z > 1$ are in $G.P.$,then $\frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z}$ are in