2024年7月5日 · The principle of mathematical induction is sometimes referred to as PMI. It is a technique that is used to prove the basic theorems in mathematics which involve the solution up to n 11 min read

Mathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true.

“Mathematical induction” is something totally different. It refers to a kind of deductive argument, a logically rigorous method of proof. It works because of how the natural numbers are constructed from set theory ; as we shall see later , PMI is “built into .”

Proof: The first step of the principle is a factual statement and the second step is a conditional one. According to this if the given statement is true for some positive integer k only then it can be concluded that the statement P(n) is valid for n = k + 1.

The Principle of Mathematical Induction (PMI) may be the least intuitive proof method available to us. Indeed, at first, PMI may feel somewhat like grabbing yourself by the seat of your pants and lifting yourself into the air.

Mathematical induction can be used to prove that an identity is valid for all integers n ≥ 1. Here is a typical example of such an identity: 1 + 2 + 3 + ⋯ + n = n(n + 1) 2. More generally, we can use mathematical induction to prove that a propositional function P(n) is true for all integers n ≥ a.

The Principle of Mathematical Induction is a technique used to prove that a mathematical statements P(n) holds for all natural numbers n = 1, 2, 3, 4, ... It helps to solve or find proof for any mathematical expression over a sequence of steps. It is proved for n = 1, n = k, and n = k + 1, and then it is said to be true for all n natural numbers.

We will give three examples of proofs that use the Principle of Mathe-matical Induction. Example 1 (The power rule). We will take the the product rule for deriva-tives as given: (fg)′ = f′g + fg′. Also assume that x′ = 1 is given. We will prove that (xn)′ = nxn−1 for all positive integers n.

Proof. Basis: When \(n = 4\) we have \(4^2 ≤ 2^4\), which is true since both numbers are \(16\). Inductive step: (In the inductive step we assume that \(k^2 ≤ 2^k \) and then show that \((k + 1)^2 ≤ 2^{k+1}\).) The inductive hypothesis tells us that \(k^2 ≤ 2^k\).

Assume that WOP is true. We want to prove PMI: Suppose .W© If and "−W a8Ð8−WÊ8 "−WÑ then WœÞ (Strategy: We will prove an equivalent statement: PMI is equivalent to If and , then "−W WÁ µÐa8ÑÐ8−WÊ8 "−WÑ which, in turn, is equivalent to If and , then "−W WÁ Ðb8ÑÐ8−W•8 "ÂWÑ