The Distribution of Partial Sum $S_n = \sum_i i \times a_i$, where $a_i$ is a random variable

Question

I think I got stuck with this question:

$\{a_i\}， i=1,2,\cdots$ $i.i.d.$ to Binary $（\pm 1， p=1/2）$

and I would like to know how to calculate the de distribution of $S_n$

where $S_n = \sum^{n}_{i=1} i \cdot a_i$

To be frank, I have no idea where I should start this question.

I think numerically, as everybody in physics departement, I would try to simulate with a Gaussian distribution. But I doubt if this is correct.....