Let $a_t\sim WN(0,\sigma_a^2)$ i.i.d (White noise) and the following ARMA(1,2) model $$(1-0.5B)Z_t=(1-1.5B+0.8B^2)a_t$$ Find the MA representation and use it to find $Var(Z_t)$
I solved the following system of equations $$\Phi(B)\Psi(B)=\Theta(B)$$ and get that $\psi_0=1$, $\psi_1=-1.0$,$\psi_2=0.3$, $\psi_k=0.5\psi_{k-1}$ for $k\geq 3$. Then the MA representation is $$Z_t=\sum_{j=0}^\infty \psi_ja_{t-j}$$ and $$Var(Z_t)=\sum_{j=0}^\infty Var(\psi_j a_{t-j})=\sigma_a^2\sum_{j=0}^\infty\psi_j^2$$ since $Cov(a_t,a_{t-j})=0$ for $j\neq 0$.
I'm a bit confused as to how I could write the result of this expression. What I get is $$Var(Z_t)=\sigma_a^2\Big(1+(-1)^2+0.3^2\sum_{j=0}^\infty0.5^{2j}\Big)=\sigma_a^2(2+\frac{0.3^2}{1-0.5^2})$$ ?