What methods exist to solve the following for $c_k,b_k$ ?
$$f(t) = \sum_k c_k{b_k}^t$$ Or phrased differently $b_k = 1+\frac{p_k}{100}$
For context... This problem has a direct economical interpretation. If you tell me your total amount of money at different discrete timestamps (the $f(t)$s) and you promise that you have all of it in some number of bank accounts which each have a constant percent of interest (and you promise you don't touch any of the money during that time).
Then can we calculate what the starting amount at each account at time $0$ (the $c_k$s) as well as the percentages of interest ($p_k$s)? Also how many timepoints would be necessary to solve it uniquely?