Sigma sums problem

Question

Could you please help me with the below result?

How do we get from here

$(1+2\sum_{j=1}^\infty\rho^j)π_0=1$

To here

$π_0=(1-\rho)/(1+\rho)$

Many thanks

Answer 1

Hint: If $\lvert \rho \rvert < 1$, then you can sum the GP : $\displaystyle \sum_{j=1}^\infty \rho^j = \dfrac{\rho}{1-\rho}$