A random variable X takes the values of 0, 1, 2, 3, ...., with probability $P(X=x)=k(x+1) \left(\frac{1}{5}\right)^x$, where k is a constant, then $P(X=0)$ is

$\frac{16}{25}$

The probability distribution of X is

$P(X=x) = k(x+1)\left(\frac{1}{5}\right)^x, x = 0,1,2,3,...$

$∴ \sum\limits^{∞}_{x=0}P(X=x)=1$

$⇒k\sum\limits^{∞}_{x=0}(x+1)\left(\frac{1}{5}\right)^x=1$

$⇒ k\begin{Bmatrix}1+2 ×\left(\frac{1}{5}\right) + 3 ×\left(\frac{1}{5}\right)^2+4 ×\left(\frac{1}{5}\right)^3+...\end{Bmatrix}=1$

$⇒ k\begin{Bmatrix} \frac{1}{1-\frac{1}{5}}+\frac{1×\frac{1}{5}}{\left(1-\frac{1}{5}\right)^2}\end{Bmatrix}=1\left[a+(a+d)r+(a+2d)r^2+... =\frac{a}{1-r}+\frac{dr}{(1-r)^2}\right]$

$⇒ k×\frac{25}{26}=1 ⇒ k=\frac{16}{25}$