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The notion of proof is central to mathematics, providing a rigorous way to establish the truth of mathematical statements. One commonly used method of proof is mathematical induction. Let’s discuss both:

  1. Introduction to Proof:
    • In mathematics, a proof is a logical argument that demonstrates the validity of a mathematical statement.
    • A proof typically consists of a series of logical steps, starting from known or assumed facts (axioms or previously proven theorems) and using deductive reasoning to derive the desired conclusion.
    • The goal of a proof is to establish that a statement (conjecture or theorem) holds true under all possible conditions or within a specified domain.
  2. Mathematical Induction:
    • Mathematical induction is a powerful technique used to prove statements about natural numbers.
    • It consists of two steps: a. Base Case: Proving that the statement holds true for some initial value, typically the smallest value of the domain (often
      or ). b. Inductive Step


      (the induction hypothesis), and then proving that it also holds true for the next value


    • If both the base case and the inductive step are valid, mathematical induction guarantees that the statement holds true for all natural numbers greater than or equal to the initial value.
    • Mathematical induction is often used to prove statements about sums, products, divisibility, inequalities, and other properties of natural numbers.
    • There are variations of mathematical induction, such as strong induction and complete induction, which involve different assumptions or methods for proving the inductive step.

Mathematical induction is a fundamental proof technique in mathematics, widely used in various branches such as number theory, combinatorics, algebra, and analysis. It provides a systematic and reliable method for establishing the truth of statements involving natural numbers.