Let $a$, $b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x ^{2}-8 ax +2 a =0$ and $q$ and $s$ are the roots of the equation $x^{2}+12 b x+6 b$ $=0$, such that $\frac{1}{ p }, \frac{1}{ q }, \frac{1}{ r }, \frac{1}{ s }$ are in A.P., then $a ^{-1}- b ^{-1}$ is equal to $……$

For $\mathrm{x} \geq 0$, the least value of $\mathrm{K}$, for which $4^{1+\mathrm{x}}+4^{1-\mathrm{x}}$, $\frac{\mathrm{K}}{2}, 16^{\mathrm{x}}+16^{-\mathrm{x}}$ are three consecutive terms of an $A.P.$ is equal to :

Let ${S_1},{S_2},……,{S_{101}}$ be the consecutive terms of an $A.P$ . If $\frac{1}{{{S_1}{S_2}}} + \frac{1}{{{S_2}{S_3}}} + …. + \frac{1}{{{S_{100}}{S_{101}}}} = \frac{1}{6}$ and ${S_1} + {S_{101}} = 50$ , then $\left| {{S_1} – {S_{101}}} \right|$ is equal to

Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic progression. If $S_{3 n}=3 S_{2 n}$, then the value of $\frac{S_{4 n}}{S_{2 n}}$ is:

If three numbers be in $G.P.$, then their logarithms will be in