If $3 + 3\alpha + 3\alpha^2 + \dots \infty = \frac{45}{8}$,then the value of $\alpha$ will be

If the sum of the first $n$ terms of the series $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \dots$ is $\frac{n(n+1)^2}{2}$ when $n$ is even,what is the sum when $n$ is odd?

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

Find the sum to $n$ terms of the series $3 \times 1^{2} + 5 \times 2^{2} + 7 \times 3^{2} + \dots$

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

If $a, x$ are real numbers and $|a| < 1, |x| < 1$,then $1 + (1+a)x + (1+a+a^2)x^2 + \dots \infty$ is equal to