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What is the mathematical idea behind recursion?

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What is the mathematical idea behind recursion?

In mathematics, many problems can be written in recursive form. For example, the sum of natural numbers can be expressed using a smaller version of the same problem.

Sum of n natural numbers = n + Sum of (n - 1) natural numbers
Expression Recursive Meaning
Sum(n) n + Sum(n - 1)
Sum(n - 1) (n - 1) + Sum(n - 2)
Sum(n - 2) (n - 2) + Sum(n - 3)