Bellman equations

The Bellman equations are named after Richard Bellman (https://en.wikipedia.org/wiki/Richard_E._Bellman), who also introduced the DP method. In DP, we can find the optimal solution of a complex problem by breaking it down into simpler, recursive sub-problems and finding their optimal solutions. For example, we can find the k-th Fibonacci number using a bottom-up dynamic method.

Although there is a strong chance that you are already familiar with this solution, we'll still include it for the sake of completeness (it's just a few sentences). We'll start with the first and second Fibonacci numbers (0 and 1). The third number is just the sum of the first two. Then, the fourth number is the sum of the third and the second, but ...

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