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

$8^{th}$ term of the series $2\sqrt 2 + \sqrt 2 + 0 + .....$ will be

If $n$ is the smallest natural number such that $n+2 n+3 n+\ldots+99 n$ is a perfect square, then the number of digits of $n^2$ is

If the sum of the series $2 + 5 + 8 + 11............$ is $60100$, then the number of terms are

The arithmetic mean of the nine numbers in the given set $\{9,99,999,...., 999999999\}$ is a $9$ digit number $N$, all whose digits are distinct. The number $N$ does not contain the digit

The sum of all those terms, of the anithmetic progression $3,8,13, \ldots \ldots .373$, which are not divisible by $3$,is equal to $.......$.