I have three sequences $a_1, a_2, \cdots, a_t$, $b_1, b_2, \cdots, b_t$, and $c_1, c_2, \cdots, c_t$ where the subscripts represent different time steps. At step $t$, $c_t = \frac{\sum_{i=1}^t b_i}{1+\sum_{i=1}^t a_i}$.
I want to remove the summation over time so that $c_t$ only depends on $c_{t-1}$, $a_t$, and $b_t$. Is it possible to do so? Can someone please help me understand this? Thank you very much.
I don't think that it is possible to do so unless the sequences with $a$ and $b$ follow a known pattern.
This might not be the best example, but think about this specific case: $c_{t-1}$ evaluates to $\frac{np}{p}$ where $p$ is a prime number and $a_t$, $b_t$ are both $=1$. In this case, you are trying to find an expression for $\frac{np+1}{p+1}$ using only $n$ and $1$ (i.e. without having $p$ at your disposal), which is not possible.