The sum of the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \dots$ to $9$ terms is

For three positive integers $p, q, r$,$x^{pq p^2} = y^{qr} = z^{p^2 r}$ and $r = pq + 1$ such that $3, 3 \log_y x, 3 \log_z y, 7 \log_x z$ are in $A$.$P$. with common difference $\frac{1}{2}$. Then $r - p - q$ is equal to

If the $n^{th}$ terms of two $A.P.$'s are $3n + 8$ and $7n + 15$,then the ratio of their $12^{th}$ terms will be:

The series of positive multiples of $3$ is divided into sets: $\{3\}, \{6, 9, 12\}, \{15, 18, 21, 24, 27\}, \ldots$. Then the sum of the elements in the $11^{\text{th}}$ set is equal to $................$

If the $n^{th}$ term of an arithmetic progression is $3n - 1$,then the sum of its first five terms is.......

Let $S_n$ denote the sum of $n$ terms of an $A.P.$ If $S_{2n} = 3S_n$,then the ratio $\frac{S_{3n}}{S_n} = $