Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a $G.P.$ be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an $A.P.$ If the $5^{\text {th }}$ term of the $G.P.$ is the $\mathrm{n}^{\text {th }}$ term of the $A.P.$, then $\mathrm{n}$ is equal to

$x + y + z = 15$ if $9,\;x,\;y,\;z,\;a$ are in $A.P.$; while $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{3}$ if $9,\;x,\;y,\;z,\;a$ are in $H.P.$, then the value of $a$ will be

If $a, b, c$ are in $GP$ and $4a, 5b, 4c$ are in $AP$ such that $a + b + c = 70$,  then value of $a^3 + b^3 + c^3$ is

Let $a, b, c > 1, a^3, b^3$ and $c^3$ be in $A.P.$, and $\log _a b$, $\log _c a$ and $\log _b c$ be in G.P. If the sum of first $20$ terms of an $A.P.$, whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then abc is equal to

If the ratio of two numbers be $9:1$, then the ratio of geometric and harmonic means between them will be

If three unequal non-zero real numbers $a,\;b,\;c$ are in $G.P.$ and $b - c,\;c - a,\;a - b$ are in $H.P.$, then the value of $a + b + c$ is independent of