Lesson plan	Lesson plan
What to create	Lesson plan
Which subject	Mathematics
What topic	Inductive Proof
What length (min)	30
What age group	College
Include homework
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Overview

This lesson will cover the concept of inductive proof in mathematics. It will begin by defining inductive proof and its importance in mathematics. Then, it will provide examples of how to construct inductive proofs and how they can be used to prove statements about integers. This lesson is designed for college-level mathematics students and will take approximately 30 minutes to complete.

Goals

• Understand the concept of inductive proof
• Learn how to construct inductive proofs
• Understand how inductive proofs can be used to prove statements about integers

Objectives

By the end of this lesson, students will be able to:

• Explain the concept of inductive proof
• Construct an inductive proof for a given statement
• Apply inductive proof to prove statements about integers

Lesson Content

Introduction to Inductive Proof

Inductive proof is a type of mathematical proof that uses examples to prove that a statement is true for all values of a given parameter. It is an important technique used in many areas of mathematics, including analysis, set theory, and number theory.

Constructing Inductive Proofs

To construct an inductive proof, one must first show that the statement is true for the smallest possible value of the parameter. This is usually the base case. Next, assume that the statement is true for some arbitrary value of the parameter, and use this assumption to prove that the statement is also true for the next value of the parameter. This is called the inductive step. Finally, use mathematical induction to show that the statement is true for all values of the parameter.

Example of Inductive Proof

Suppose we want to prove that the sum of the first n even integers is n(n + 1).

• Base case: When n = 1, the sum of the first even integer (2) is 2 = 1(1 + 1). Thus, the statement is true for n = 1.
• Inductive step: Assume that the statement is true for n = k. Then the sum of the first k even integers is k(k + 1). We want to prove that the statement is also true for n = k + 1. Notice that the (k + 1)th even integer is (2k + 2). Thus, the sum of the first k + 1 even integers is (k + 1)(k + 2), which simplifies to k(k + 1) + 2(k + 1). Using our assumption for n = k, we can replace k(k + 1) with its equivalent expression, giving us (k + 1)(k + 2). Thus, the statement is true for n = k + 1.
• Using mathematical induction, we can say that the statement is true for all values of n.

Applications of Inductive Proof

Inductive proofs are often used to prove statements about integers. For example, they can be used to prove that a given formula works for all integers, or that a certain property holds for all positive integers.

Conclusion

Inductive proofs are a powerful tool in mathematics. They allow us to prove statements about integers using examples and mathematical induction. By constructing inductive proofs, we can prove that a statement is true for all values of a given parameter.

Homework

1. Prove that the sum of the first n odd integers is n^2.
2. Prove that the sum of the first n natural numbers is (n(n + 1))/2.
3. Use an inductive proof to show that 11^n - 6^n is always divisible by 5.