A discrete random variable $X$ of values in $\mathbb N$ verifies the property that $P(X=k)=\cfrac 23 (k+1)P(X=k+1)$ What is the distribution of $X$?
I found that $P(X\ge 0)=\sum_{k=0}^\infty P(X=k)=\sum_{k=0}^\infty\cfrac 23 (k+1)P(X=k+1)=\cfrac 23\sum_{k=1}^\infty kP(X=k)=\cfrac 23\text E(X)=1$ $\ \ \ \ \ \ \ \ \text E(X) = 1.5$
I also found that $P(X=k)=\cfrac{3^k}{2^k\cdot k!}\cdot P(X=0)$ That is the only thing I could get out of the given property, I couldn't find the expression for $P(X=k)$ which is the actual question.