In a problem I'm working on, I develop an expression: $$\sum_{i=1}^NB_i =K_4 \frac{\sum_{i=1}^NM_i}{K_2+\sum_{i=1}^NM_i}$$ What I really want is an expression for an individual $B_i$. Through various means, I can demonstrate that $$B_i =K_4 \frac{M_i}{K_2+\sum_{i=1}^NM_i}$$ Arriving at this result took a considerable effort because I was avoiding the "naive" idea to simply remove the summation signs on $B_i$ and $M_i$. However, that naive approach does produce the correct result in this instance.
My Question: Is there a way to know if the "naive" approach is valid when equating finite summations with the same number of objects?