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 the $p^{th}$ term of an arithmetic progression is $q$ and the $q^{th}$ term is $p$,then its $r^{th}$ term is:

The sum of all natural numbers between $1$ and $100$ which are multiples of $3$ is

If $n$ arithmetic means are inserted between $a$ and $100$ such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$,then the value of $n$ is

Suppose $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ is an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of the first nine terms of the progression is $5:17$ and $110 < a_{15} < 120$,then the sum of the first ten terms of the progression is equal to -

Insert five numbers between $8$ and $26$ such that the resulting sequence is an $A.P.$