If the arithmetic mean and geometric mean of the $p^{\text{th}}$ and $q^{\text{th}}$ terms of the sequence $-16, 8, -4, 2, \ldots$ satisfy the equation $4x^{2}-9x+5=0$,then $p+q$ is equal to ..... .

If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0$ and $\gamma, \delta$ are the roots of $x^2 - 12x + b = 0$,and the numbers $\alpha, \beta, \gamma, \delta$ (in order) form an increasing $G.P.$,then:

If $x, y, z$ are in $G.P.$ and $a^x = b^y = c^z$,then

All terms of a geometric progression are positive. If each term is equal to the sum of the two terms following it,then the common ratio of this progression is:

For a $G.P.$,if $(m+n)^{\text{th}}$ term is $p$ and $(m-n)^{\text{th}}$ term is $q$,then the $m^{\text{th}}$ term is $.........$

Consider two $G$.Ps. $2, 2^{2}, 2^{3}, \ldots$ and $4, 4^{2}, 4^{3}, \ldots$ of $60$ and $n$ terms respectively. If the geometric mean of all the $60+n$ terms is $(2)^{\frac{225}{8}}$,then $\sum_{k=1}^{n} k(n-k)$ is equal to.