Find the sum of the series $6 + 66 + 666 + \dots$ up to $n$ terms.

If $\sum_{r=1}^{n}r^3 - \sum_{p=1}^{n}\sum_{m=1}^{p}\sum_{r=1}^{m}1 = 80$,then the possible value of $n$ is:

Find the sum of $n$ terms of the series $1 + 6 + 13 + 22 + 33 + \dots$

If $b$ is the first term of an infinite $G.P.$ whose sum is $5$,then $b$ lies in the interval

Let $S$ be the infinite sum given by $S = \sum_{n=0}^{\infty} \frac{a_n}{10^{2n}}$,where $(a_n)_{n \geq 0}$ is a sequence defined by $a_0 = 1, a_1 = 1$ and $a_j = 20a_{j-1} - 108a_{j-2}$ for $j \geq 2$. If $S$ is expressed in the form $\frac{a}{b}$,where $a$ and $b$ are coprime positive integers,then $a$ equals:

The value of $(0.05)^{\log_{\sqrt{20}}(0.1 + 0.01 + 0.001 + \dots)}$ is