Question

summation 1+2(1/5)+3(1/5)^2+4(1/5)^3 ........

Answer 1

$\frac{25}{16}$

Answer 2

We have the series

$$S = 1 + 2\left(\frac{1}{5}\right) + 3\left(\frac{1}{5}\right)^2 + 4\left(\frac{1}{5}\right)^3 + \cdots$$

This can be written as:

$$S = \sum_{n=1}^{\infty} n\left(\frac{1}{5}\right)^{n-1}$$

For $|r| < 1$, the formula for the sum is:

$$\sum_{n=1}^{\infty} n r^{n-1} = \frac{1}{(1-r)^2}$$

Here, $r = \frac{1}{5}$. Hence,

$$S = \frac{1}{\left(1-\frac{1}{5}\right)^2} = \frac{1}{\left(\frac{4}{5}\right)^2} = \frac{1}{\frac{16}{25}} = \frac{25}{16}$$