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?
Best Answer
Set $$ Y=K_4\frac{\sum_{i=1}^NM_i}{K_2+\sum_{i=1}^NM_i} $$ Now you can choose $$ B_1=Y,\quad B_i=0, 1<i\le N $$ or $$ B_n=Y,\quad B_i=0, 1\le i<N $$ or, of course, infinitely many other ways to have the same sum on the left-hand side as on the right-hand side (unless $N=1$, of course).
Without other relations between $B_i$ and $M_i$ there's no way to remove the summation from both sides.