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

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

Let $S_n$ be the sum of all integers $k$ such that $2^n < k < 2^{n+1}$,for $n \geq 1$. Then,$9$ divides $S_n$ if and only if

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an arithmetic sequence are $a$,$b$,and $c$ respectively,then the value of $[a(q - r) + b(r - p) + c(p - q)]$ is:

Suppose that all the terms of an arithmetic progression $(A.P.)$ are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6:11$ and the seventh term lies between $130$ and $140$,then the common difference of this $A.P.$ is

If $S_1, S_2, S_3, \dots, S_m$ are the sums of $n$ terms of $m$ $A.P.s$ whose first terms are $1, 2, 3, \dots, m$ and common differences are $1, 3, 5, \dots, 2m - 1$ respectively,then $S_1 + S_2 + S_3 + \dots + S_m = $