11.4 Mathematical Induction Objective: To prove a statement is true for all positive integers.

What is mathematical induction? It is taking a statement and proving it is true for all positive integers.

Mathematical Induction If we are given a series (summation of an infinite sequence), we want to be able to prove that the statement is true for all values of n. Ex 1: Sn = 1 + 2 + 3 + … + n = n(n+1) 2 …What are the steps to the proof?

Proof by Mathematical Induction Let Sn be a statement involving positive integer n. Step 1: Show that S1 is true. substitute n=1 and show it is true Step 2: Assume Sk is true. write the statement “Assume (substitute n=k) is true.” Step 3: Show Sk+1 is true. start with statement from step 2 and add the next term (the “k+1” term) Replace the first part with the assumption from step 2 Manipulate the LEFT SIDE ONLY to become the same as the right side

Back to Ex1. Use mathematical induction to prove that 1 + 2 + 3 + …+ n = n(n+1)/2 for all n. Step 1: Show S1 is true. Step 2: Assume Sk Step 3: Prove Sk+1

Ex2. Use mathematical induction to prove: Step 1: Show S1 is true. Step 2: Assume Sk Step 3: Prove Sk+1

Ex3. Use mathematical induction to prove: Step 1: Show S1 is true. Step 2: Assume Sk Step 3: Prove Sk+1

You try. Use mathematical induction to prove: Step 1: Show S1 is true. Step 2: Assume Sk Step 3: Prove Sk+1