When faced with $N$ beads on a string, I found the following equation
$$-\omega^2A_p+2\omega_0^2A_p-\omega_0^2(A_{p+1}+A_{p-1})=0$$ Where $p=1,2,\dots,N$ and $A_0=A_{N+1}=0$
I know I can't solve for $\omega$ and the $N$ $A$'s since I only have $N$ equations, but I am interested to solve for $\omega$ and the ratios $\frac{A_{p+1}}{A_p}$ for $p=1,2,\dots,N-1$, leaving me with only $N$ variables. Can somebody guide me through an intelligent way to go through this problem?
Your equations are homogeneous (or seem to be), so you can take any solution for the $A$s and multiply by a constant to get a new solution. So simply set $A_1 = 1$ and try to solve. If that doesn't work, set $A_2 = 1$ and try, and so on. If none of these has a solution, then the only solution is one with all the $A_i = 0$.