If $64, 27, 36$ are the $P^{\text{th}}$, $Q^{\text{th}}$, and $R^{\text{th}}$ terms of a $GP$, then $P+2Q$ is equal to

Let $729, 81, 9, 1, \dots$ be a sequence and $P_{n}$ denote the product of the first $n$ terms of this sequence. If $2\sum_{n=1}^{40}(P_{n})^{\frac{1}{n}}=\frac{3^{\alpha}-1}{3^{\beta}}$ and $\gcd(\alpha,\beta)=1$, then $\alpha+\beta$ is equal to

Let $S_1$ be the sum of areas of the squares whose sides are parallel to the 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

If the first and the $n^{\text{th}}$ term of a $G.P.$ are $a$ and $b$ respectively,and if $P$ is the product of $n$ terms,prove that $P^{2} = (ab)^{n}$.

Find the geometric mean of the first $n$ natural numbers.

If the roots of the equation $3x^3-26x^2+52x-24=0$ are in geometric progression,then the sum of two of its roots is