1. 11.4 Mathematical Induction
Objective: To prove a statement
is true for all positive integers.

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

3. 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 = … + n = n(n+1) 2
…What are the steps to the proof?

4. 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

5. 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

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

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

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