This chapter shows some recursive and other functional solutions to simple problems, as well as some ways we can test and improve these solutions. Two very common programming topics that demonstrate functional programming are the factorial and Fibonacci mathematical series—largely because they’re so easily described using recursive means.^{[20]}

The *factorial* of a given positive number is the product of all the integers from 1 to that number, so factorial(3) = 3 x 2 x 1, factorial(5) = 5 x 4 x 3 x 2 x 1, and so on. This can be expressed generally as:

factorial(

x) =xx (x– 1) x (x– 2) ...

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