## Find constants $b_i$ such that $\sum_{n=1}^{N} a_n (\sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$

I am trying to find the real constants $$b_i$$ such the $$\displaystyle \sum_{n=1}^{N} a_n \left(\displaystyle \sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))\right)=0$$ where I am given that $$\displaystyle \sum_{n=1}^{N} a_n r_n=0$$ for every natural $$N$$ and $$s(i)$$ is certain function of $$i$$ (I have the explicit form for $$s(i)$$ as well in terms of a series).

Note that $$t$$ is at our disposal. I tried to manipulate my expression for $$t=3$$ to find such constants $$b_1,b_2,b_3$$ unsuccessfully. I also want to mention that there is no guarantee for the existence of such constants except $$b_i=0$$ for all $$i$$.