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Mathematics LibreTexts

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.