The sum of the series $1 + 3x + 6x^2 + 10x^3 + \dots \infty$ is

The value of $\frac{1 \times 2^2 + 2 \times 3^2 + \ldots + 100 \times 101^2}{1^2 \times 2 + 2^2 \times 3 + \ldots + 100^2 \times 101}$ is

Let $\alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \infty$ and $\beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty$. Then the value of $(0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_{5}(\beta)}$ is equal to:

The sum of $n$ terms of the series $1^{3}+3^{3}+5^{3}+7^{3}+\ldots$ is

The sum of an infinite $G.P.$ with common ratio $r$ can be found:

$2.5 + 5.9 + 8.13 + 11.17 + \ldots$ to $10$ terms $=$