Does $a_1\sqrt{1}+a_2\sqrt{2}+...+a_n\sqrt{n}=0$ have no solutions over the rationals?

Question

I had recently met a question that asked to show that $a\sqrt{2}+b\sqrt{3}+c=0$ has no solutions over the rationals (except for $0$) for $a,b$ and $c$. To show this I moved $c$ to the other side then squared both sides. Then the equation re-arranges into $\sqrt{6}=\frac{c^2-2a^2-3b^2}{2ab}$, which is a contradiction as one side is irrational and the other side is rational. I was wondering if I could extend this to $n$ terms, i.e does $a_1\sqrt{1}+a_2\sqrt{2}+...+a_n\sqrt{n}=0$ have any rational solutions for any $a_1,a_2,...,a_n$ and how would I prove that if it did? This time I can't just square both sides because there are too many terms to isolate so I was wondering if anyone could help (dis)prove this statement that would be great.