If the ${p^{th}}$,${q^{th}}$ and ${r^{th}}$ term of a $G.P.$ are $a,\;b,\;c$ respectively, then ${a^{q - r}}{b^{r - p}}{c^{p - q}}$ is equal to

If $a,\;b,\;c$ are in $A.P.$, $b,\;c,\;d$ are in $G.P.$ and $c,\;d,\;e$ are in $H.P.$, then $a,\;c,\;e$ are in

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

How many terms of $G.P.$ $3,3^{2}, 3^{3}$... are needed to give the sum $120 ?$

If the ${10^{th}}$ term of a geometric progression is $9$ and ${4^{th}}$ term is $4$, then its ${7^{th}}$ term is

Let $S_1$ be the sum of areas of the squares whose sides are parallel to coordinate axes. Let $S_2$ be the sum of areas of the slanted squares as shown in the figure. Then, $\frac{S_1}{S_2}$ is equal to