Answer
$$a_n = -\frac{(-1)^(n-1)n^2}{n+1}.$$
Work Step by Step
Our first term is $a_1 = \frac{1}{2}$. The sequence alternates, so we have a factor of $(-1)^n$. We observe that we are also multiplying by $\frac{n^2}{n+1}$ each term. Since $a_1 = \frac{1}{2}$, we must have
$$a_n = -\frac{(-1)^nn^2}{n+1}.$$
