Intro to Mathematical Induction

Introduction

This tutorial provides a comprehensive guide to understanding and applying mathematical induction, a fundamental proof technique in mathematics. By following these steps, you will learn how to prove statements indexed by positive integers, including a classic example involving the sum of the first n integers.

Step 1: Establish the Basis Case

The first step in mathematical induction is to verify that the statement holds true for the initial value, usually when n = 1.

• Identify the claim you want to prove.
• Substitute n = 1 into the claim.
• Show that the result is true.

Example: If your claim is that the sum of the first n integers equals n(n + 1)/2, you would check:

• For n = 1: [ 1 = \frac{1(1 + 1)}{2} \implies 1 = 1 ]
• Thus, the basis case is true.

Step 2: Assume True for the k-th Level

Next, you make the induction assumption, which is the hypothesis that the statement holds true for some arbitrary positive integer k.

• Clearly state your assumption.
• This serves as the foundation for proving the next step.

Example: Assume the statement is true for n = k: [ 1 + 2 + ... + k = \frac{k(k + 1)}{2} ]

Step 3: Prove the (k + 1)-th Level

Using the induction assumption, you will now show that if the statement is true for n = k, then it must also be true for n = k + 1.

• Start from the induction assumption.
• Manipulate the equation to include the (k + 1) term.
• Show that the resulting expression matches the form required for n = k + 1.

Example: Starting from the assumption: [ 1 + 2 + ... + k = \frac{k(k + 1)}{2} ] Add (k + 1) to both sides: [ 1 + 2 + ... + k + (k + 1) = \frac{k(k + 1)}{2} + (k + 1) ] Combine the right-hand side: [ = \frac{k(k + 1) + 2(k + 1)}{2} = \frac{(k + 1)(k + 2)}{2} ] This shows that the claim holds for n = k + 1.

Conclusion

In this tutorial, you learned the three essential steps of mathematical induction: establishing the basis case, making an induction assumption, and proving the next level. By mastering these steps, you can tackle a variety of mathematical proofs, including demonstrating the formula for the sum of the first n integers. For continued learning, consider practicing more induction problems to solidify your understanding.