$X_1 ,\ldots, X_n$ are independent with distribution $X_i=\pm \sqrt{i}$ with probability $0.5$. Get the asymptotic distribution of the mean $\bar{X}$.

Question

$X_1 ,\ldots, X_n$ are independent with distribution $X_i=\pm \sqrt{i}$ with probability $0.5$. Get the asymptotic distribution of the mean $\bar{X}$

Seems very easy. But I used mgf and find it's hard to calculate. To apply CLT I can make it to something like $$\sqrt{n}\frac{\sum_{i=1}^{n}X_i/\sqrt{n}}{n}$$But every term in the sum is related to $n$... Any ideas? Thanks!