The arithmetic mean of the first $n$ natural numbers is:

The number of terms common between the two series $2 + 5 + 8 + \dots$ up to $50$ terms and the series $3 + 5 + 7 + 9 + \dots$ up to $60$ terms is:

Let $S_n = 1 + q + q^2 + ..... + q^n$ and $T_n = 1 + \left( \frac{q + 1}{2} \right) + \left( \frac{q + 1}{2} \right)^2 + ...... + \left( \frac{q + 1}{2} \right)^n$ where $q$ is a real number and $q \neq 1$. If $^{101}C_1 + ^{101}C_2 \cdot S_1 + ...... + ^{101}C_{101} \cdot S_{100} = \alpha \cdot T_{100}$,then $\alpha$ is equal to

Let $S_1, S_2, \dots$ be squares such that for each $n \ge 1$,the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10 \text{ cm}$,then for which of the following values of $n$ is the area of $S_n$ less than $1 \text{ cm}^2$?

Determine the sum of the first $35$ terms of an $A.P.$ if $t_{2}=2$ and $t_{7}=22$.

If $a_1, a_2, a_3, ..., a_n$ are in $A.P.$,where $a_i > 0$ for all $i$,then the value of $\frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + ... + \frac{1}{\sqrt{a_{n-1}} + \sqrt{a_n}} = $