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

If the sum of infinite terms of a $G.P.$ is $3$ and the sum of the squares of its terms is $3$,then its first term and common ratio are:

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

For a sequence $(t_n)$,if $S_n = 5(2^n - 1)$,then $t_n = \ldots$

If we insert two numbers between $\sqrt{2}$ and $4$ so that the resulting sequence is in $G.P.$, then the inserted numbers in the order are

Consider the quadratic equation $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2) = 0, n \in R$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the $G$.$P$.,whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is: