The mean of the squares of the first $n$ natural numbers is

For $x \in \mathbb{R}$,let $[x]$ denote the greatest integer $\le x$. Find the sum of the series $\left[ -\frac{1}{3} \right] + \left[ -\frac{1}{3} - \frac{1}{100} \right] + \left[ -\frac{1}{3} - \frac{2}{100} \right] + \dots + \left[ -\frac{1}{3} - \frac{99}{100} \right]$.

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:

If $\sum\limits_{i = 1}^n i = \frac{n(n + 1)}{2}$,then $\sum\limits_{i = 1}^n (3i - 2) = $

The sum of the first $n$ terms of the series $\frac{3}{5}+\frac{21}{25}+\frac{117}{125}+\ldots$ is

The sum of the series $3 + 33 + 333 + \dots$ to $n$ terms is