The sum of $n$ terms of the series $12 + 16 + 24 + 40 + \dots$ is

If $\alpha, \beta$ are natural numbers such that $100^{\alpha} - 199\beta = (100)(100) + (99)(101) + (98)(102) + \ldots + (1)(199)$,then the slope of the line passing through $(\alpha, \beta)$ and the origin is:

$1^2+\left(1^2+2^2\right)+\left(1^2+2^2+3^2\right)+\ldots+\left(1^2+2^2+\ldots+n^2\right)=$

The $99^{th}$ term of the series $2 + 7 + 14 + 23 + 34 + \dots$ is

The sum of the first $10$ terms of the series $9+99+999+\ldots$ is

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: