The number of terms common to the two arithmetic progressions $3, 7, 11, \ldots, 407$ and $2, 9, 16, \ldots, 709$ is

Let $S_n$ and $s_n$ denote the sum of the first $n$ terms of two different $A.P.$ for which $\frac{s_n}{S_n} = \frac{3n - 13}{7n + 13}$. Find the ratio $\frac{s_n}{S_{2n}}$.

If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is

The value of $(2 \cdot {}^{1}P_{0} - 3 \cdot {}^{2}P_{1} + 4 \cdot {}^{3}P_{2} - \dots$ up to $51^{\text{th}}$ term) + $(1! - 2! + 3! - \dots$ up to $51^{\text{th}}$ term) is equal to

In a $H.P.$,the $p^{th}$ term is $q$ and the $q^{th}$ term is $p$. Then the $pq^{th}$ term is:

The product $(32)(32)^{1/6}(32)^{1/36} \dots \infty$ is