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# 4: Mathematical Induction

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Mathematical induction is a mathematical proof technique that is used to prove that a property $$P(n)$$ holds for every natural number $$n$$, i.e. for n = 0, 1, 2, 3, and so on.

• 4.1: The Principle of Mathematical Induction
In this section, we will learn a new proof technique, called mathematical induction, that is often used to prove statements of the form (∀n∈N)(P(n))
• 4.2: Other Forms of Mathematical Induction
• 4.3: Induction and Recursion
In a proof by mathematical induction, we “start with a first step” and then prove that we can always go from one step to the next step. We can use this same idea to define a sequence as well. We can think of a sequence as an infinite list of numbers that are indexed by the natural numbers. Another way to define a sequence is to give a specific definition of the first term and then state how to determine the next term in terms of previous terms; this process is known as definition by recursion.
• 4.S: Mathematical Induction (Summary)

Thumbnail: Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Image used wtih permission (CC BY-SA 3.0; Pokipsy76).

4: Mathematical Induction is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Ted Sundstrom (ScholarWorks @Grand Valley State University) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.