1. Section 8.4 Mathematical Induction

2. Mathematical Induction In this section we are going to perform a type of mathematical proof called mathematical induction. This is used to show that a formula used to find the sum of a series is true. This type of proof requires a 3-step approach. Step 1: Begins with the word SHOW Step 2: Begins with the word ASSUME Step 3: Begins with the work PROVE

3. Before we begin, one needs to be able to find S k+1 based on S k. Example: S k = 3k + 2, find S k+1 Substitute k + 1 in for k found in S k.Solution:

4. Mathematical Induction Now you try: Given S k, find S k+1 for each of the following:

5. Mathematical Induction Now, let’s begin proving by mathematical induction. Step 1: Show S 1 = 1 is true. This means to substitute 1 into the formula part and show that you get the first term of the series which would be the sum of the first term.

6. Mathematical Induction Step 2: Assume Simply replace n in the Prove statement with k.

7. Mathematical Induction Step 3: Prove Go back to the Assume Statement add k + 1 as the next term of the sequence and replace all k’s in the sum formula with k + 1. Then simplify the formula.

8. Mathematical Induction Do the Proof: Write the first part of the prove statement Remove 1 + 2 + 3 + 4 + … + k and replace it with the “formula piece” from the Assume Statement. Use your algebra skills to simplify.

9. Mathematical Induction Let’s work this one together: Make sure you get it in your notes to assist with the homework!

10. Mathematical Induction Now you try this one:

11. Mathematical Induction What you should know: How to do a mathematical prove using mathematical induction!