If $1, \log_9(3^{1-x} + 2), \log_3(4 \cdot 3^x - 1)$ are in $A.P.$, then $x$ equals:

The ratio of the $7^{th}$ to the $3^{rd}$ term of an $A.P.$ is $12:5$. Find the ratio of the $13^{th}$ to the $4^{th}$ term.

Sum of the series $1 \cdot 2015 + 2 \cdot 2014 + 3 \cdot 2013 + \dots + 2015 \cdot 1$ is equal to :-

$1 + \frac{1^3 + 2^3}{1 + 2} + \frac{1^3 + 2^3 + 3^3}{1 + 2 + 3} + \dots + \frac{1^3 + 2^3 + 3^3 + \dots + 15^3}{1 + 2 + 3 + \dots + 15} - \frac{1}{2}(1 + 2 + 3 + \dots + 15)$ is equal to

Let ${T_r}$ be the ${r^{th}}$ term of an $A.P.$ for $r = 1, 2, 3, ....$. If for some positive integers $m, n$ we have ${T_m} = \frac{1}{n}$ and ${T_n} = \frac{1}{m}$,then ${T_{mn}}$ equals

If the sum of three terms of a $G.P.$ is $19$ and their product is $216$,then the common ratio of the series is