If $ \sum\limits_{n=1}^{N} a_n r_n=0$, what can we say about $ \sum\limits_{n=1}^{N} a_n (r_n)^2=0$

Question

If $\displaystyle \sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_n\in \mathbb{R}$ and $r_n\in(0,1]$, can we conclude anything significant about $\displaystyle \sum_{n=1}^{N} a_n (r_n)^2,\displaystyle \sum_{n=1}^{N} a_n (r_n)^3.. $?

EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $\sum_{n=1}^{N} a_n (\sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)