The context doesn't really matter here. I'm stuck while doing an assignment. I have the following recursive formula: $$\pi_i = 2 \rho^i \pi_0$$ for $i=1,2,3,4$.
Now I know that $\sum^4_{i=1} \pi_i =1$. In the solution it says that from this it follows that $$\pi_0 (1 + \sum^4_{i=1} \rho^i) = 1$$ or $$\pi_0 \left(2 \frac{1-\rho^5}{1-\rho}\right) =1 .$$ i.e. $\pi_0 = \frac{1-\rho}{1+\rho-2\rho^5}$
I don't see how to obtain this result. What I have tried: $$\pi_i = 2 \rho^i \pi_0$$ $$\sum_{i=0}^4 \pi_i = \sum_{i=0}^4 2 \rho^i \pi_0$$ $$1 = 2\pi_0 \left( \sum_{i=0}^4 \rho^i\right) = 2\pi_0 \left( \sum_{i=0}^\infty \rho^i - \sum_{i=5}^\infty \rho^i \right) = 2\pi_0 \left( \sum_{i=0}^\infty \rho^i - \sum_{i=0}^\infty \rho^5 \rho^i \right) = 2\pi_0 \left( \frac{1}{1-\rho} - \frac{\rho^5}{1-\rho}\right)$$
But from this is then follows that $\pi_0 = \frac{1-\rho}{2-2\rho^5}$. I'm not sure what I do wrong.