How to show that $\gcd(a_1,a_2,\cdots,a_k) = 1$ implies that there exist a non-negative solution to $\sum_{i=1}^{n}a_ix_i = n$ for large $n.$

A simple approach to the title:

Thanks to Bezout, there is some (not necessarily) integer combination of the $a_i$s that makes $1$. Add a large enough multiple of $a_1$ to each coefficient to make it non-negative and call the sum $k$; we have $k\equiv 1 \pmod{a_1}$.

Now any number $\ge ka_1$ is a non-negative combination of the $a_i$s. Namely, let $n$ be the desired number, and let $m=n\bmod a_1$. Then $mk$ is a positive combination, and we can add some multiple of $a_1$ to make it $n$ instead, since $mk\equiv n\pmod{a_1}$ and $mk<n$ because $m<a_1$.