It suffices to show \(\displaystyle \frac{1}{n}\leq a_1^2+\ldots+a_n^2\) since \(\displaystyle \frac{1}{n^2}\leq \frac{1}{n}\).

How about using calculus? Lagrange Multipliers might do the trick. Optimize \(\displaystyle f(a_1,\ldots, a_n)=a_1^2+\ldots+a_2^2\) subject to \(\displaystyle g(a_1,\ldots, a_n)=a_1+\ldots+a_n=1\).