Let be two sequences of real numbers $a = [a_1, \ldots, a_n]$ and $b = [b_1,\ldots , b_n]$. Is it possible to represent the fraction:
$$ F(a,b) = \frac{ \sum_{i=1}^{n} a_i }{ \sum_{j=1}^{n} b_j } $$
As a sum of terms such as each one is defined only by the previous numbers in the sequences?
$$ F(a,b) = \sum_{i=1}^{n} f(i) $$
with:
$$ f(i) = f( a_1, ..., a_i, b_1, ..., b_i ) $$
while avoiding trivial cases ($f(n)=F(a,b)$, $f(i) = 0 \forall i \in [1,n[$, $f(i) = \text{constant}$, etc.).