If $\log 2, \log (2^n - 1)$ and $\log (2^n + 3)$ are in $A.P.$,then $n =$

The common difference of the $A.P.$ $b_{1}, b_{2}, \ldots, b_{m}$ is $2$ more than the common difference of $A.P.$ $a_{1}, a_{2}, \ldots, a_{n}$. If $a_{40} = -159$,$a_{100} = -399$ and $b_{100} = a_{70}$,then $b_{1}$ is equal to:

If $a_1, a_2, a_3, \dots, a_n$ are in $A.P.$ and $a_1 + a_4 + a_7 + \dots + a_{16} = 114$,then $a_1 + a_6 + a_{11} + a_{16}$ is equal to

The sum of all integers between $1$ and $100$ (both inclusive) which are divisible by $5$ or $13$ is

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

If $1, \log_{y}x, \log_{z}y, -15\log_{x}z$ are in $A.P.$,then