If $a,\;b,\;c$ are ${p^{th}},\;{q^{th}}$ and ${r^{th}}$ terms of a $G.P.$, then ${\left( {\frac{c}{b}} \right)^p}{\left( {\frac{b}{a}} \right)^r}{\left( {\frac{a}{c}} \right)^q}$ is equal to

If $\frac{{x + y}}{2},\;y,\;\frac{{y + z}}{2}$ are in $H.P.$, then $x,\;y,\;z$ are in

If $\frac{a+b x}{a-b x}=\frac{b+c x}{b-c x}=\frac{c+d x}{c-d x}(x \neq 0),$ then show that $a, b, c$ and $d$ are in $G.P.$

A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter. Find the amount spent on the postage when $8^{\text {th }}$ set of letter is mailed.

If the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a $G.P.$ are $a, b$ and $c,$ respectively. Prove that

$a^{q-r} b^{r-p} c^{p-q}=1$

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