By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Math Induction By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: March 30, 2011
Before we begin…… If Pk = 1 + 5 + 9 + . . . + (4k-3) find Pk+1. Pk+1 = 1 + 5 + 9 + . . . + (4k-3) + (4(k+1) – 3) Pk+1 = 1 + 5 + 9 + . . . + (4k-3) + (4k+1)
Before we begin…… If Pk = k2(k+1)2 find Pk+1. Pk+1 = (k+1)2(k+1+1)2 = (k+1)2(k+2)2
The Principle of Math Induction Let Pn be a statement involving the positive integer n. If 1. P1 is true the truth of Pk implies the truth of Pk+1 for every positive k then Pn must be true for all positive integers n. To apply the Principle of Math Induction, you need to be able to determine the statement Pk+1 for a given statement Pk. To determine Pk+1, substitute k+1 for k in the statement Pk. Direct Quote: Precalculus with Limits – A Graphin Approach by Larson, Hostetler & Edwards © 2005 by Houghton Mifflin Company
Induction – A three step process Step 1: Show P1 is true APPROACH THE Step 2: Assume Pk is true Step 3: Prove Pk+1 is true A “Bivinism”
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Prove: Show P1: Assume Pk: Prove Pk+1: Q.E.D.
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 1
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 2
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 3
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 4
Prove: Show P1: Assume Pk: Prove Pk+1: Q.E.D.
Prove: Show P1: Assume Pk: Prove Pk+1:
Worksheet Exercise # 5
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 6
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 7
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 8
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.
Worksheet Exercise # 9
Prove: Show P1: Assume Pk: Prove Pk+1:
We need a common denominator to add the fractions. Prove Pk+1: We need a common denominator to add the fractions.
Prove Pk+1: We see that a factor of (k+1) is in the denominator on the left that is not on the right. Maybe the numerator has a (k+1) factor that will cancel with it. We will use synthetic division to check. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 14 19 8 -1 -3 -11 -8 11 Q.E.D.
Worksheet Exercise # 10
Prove: Show P1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D.