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 the positive numbers $a_1, a_2, a_3, a_4$ and $a_5$ be in a $G$.$P$. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively,where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$,then $m+n$ is equal to $.........$.

If the first term of a $G.P.$ is $5$ and the common ratio is $-5$,then which term is $3125$ (in $^{th}$)?

If $x = 1 + a + a^2 + \dots \infty$ $(a < 1)$ and $y = 1 + b + b^2 + \dots \infty$ $(b < 1)$,then the value of $1 + ab + a^2b^2 + \dots \infty$ is

If the sum of three terms of a $G.P.$ is $19$ and their product is $216$,then the common ratio of the series is

If every term of a $G.P.$ with positive terms is the sum of its two previous terms,then the common ratio of the series is