Find the geometric mean between $\frac{3}{2}$ and $\frac{27}{2}$.

The sum of all the digits of the numbers from $1$ to $100$ is

If $a_1, a_2, a_3, . . . , a_n, . . .$ are in $A.P.$ such that $a_4 - a_7 + a_{10} = m$,then the sum of the first $13$ terms of this $A.P.$ is .............. $m$.

The minimum value of ${\left( {\frac{3}{a} - 1} \right)^2} + {\left( {\frac{a}{b} - 1} \right)^2} + {\left( {\frac{b}{c} - 1} \right)^2} + {\left( {3c - 1} \right)^2}$ where $0 < a, b, c \leqslant 9$,is $p - q\sqrt{r}$; $p, q, r \in I$ and $q, r$ are coprime,then $(p + q + r)$ is equal to

If $a_1, a_2, a_3, \dots$ are in $A.P.$ such that $a_1 + a_7 + a_{16} = 40$, then the sum of the first $15$ terms of this $A.P.$ is

The sum to infinity of the following series $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{2^3} + \frac{1}{3^3} + \dots$ will be