How to calculate autocorrelation at certain lag?

Question

I started learning time series analysis and I have one small "home task". $$ X_t = Z_t + 0.5Z_{t-1} + 0.5Z_{t-2}, \sigma^2=1 \\ Find: \gamma(0), \gamma(2) \text{ and autocorrelation function at lag 2, acf(2)} $$ Where $\gamma$ is autocovariance and $ Z_t \sim iid(0,\sigma^2) $.
I know that at lag 0: $$ \gamma(0) = \sigma^2 \sum_{i=0}^{q-k} \beta_i^2 $$ and in the general form: $$ \gamma(k) = \sigma^2 \sum_{i=0}^{q-k}\beta_i\beta_{i+k} $$ I am computing $ \gamma(0)$ and $\gamma(2) $ as: $$ \gamma(0) = 1^2 + 0.5^2 + 0.5^2 = 1.5 \\ \gamma(2) = 1 * 0.5 = 0.5 $$ I know that autocorrelation function is: $$ \rho(2) = \frac{\gamma(2)}{\gamma(0)} $$ I think, that autocorrelation function at lag 2 must be then $ \frac{0.5}{1.5} $. Am I doing something wrong? The automated test says that this is wrong. Can anyone please explain this to me?
[Edit] The calculations were right. There was a problem with the automatic test system