Find the sum of the numbers of three digits divisible by $7$.

If ${a_1}, {a_2}, \dots, {a_{n+1}}$ are in $A.P.$,then $\frac{1}{{{a_1}{a_2}}} + \frac{1}{{{a_2}{a_3}}} + \dots + \frac{1}{{{a_n}{a_{n+1}}}}$ is

If the sum of $n$ terms of a $G.P.$ is $255$,the $n^{th}$ term is $128$,and the common ratio is $2$,then the first term will be:

The sum of the squares of the first $n$ natural numbers exceeds their sum by $330$. Then,$n = $

If the $2^{nd}, 5^{th}, \text{and } 9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$, then the common ratio of this $G.P.$ is:

Let $a_1, a_2, a_3, \dots, a_{100}$ be positive real numbers and $S_k$ be the sum of products of $a_1, a_2, \dots, a_{100}$ taken $k$ at a time. If $S_{98} S_2 \ge \lambda (a_1 a_2 \dots a_{100})$,then $\lambda$ is