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$ .