by: jeffrey bivin lake zurich high school
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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)TRANSCRIPT
By: Jeffrey Bivin Lake Zurich High School
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Math Induction By:Jeffrey Bivin Lake Zurich High School Last
Updated:March 30, 2011 Before we begin If Pk = 1 + 5 + 9 + . . . +
(4k-3) find Pk+1.
Pk+1 = (4k-3) + (4(k+1) 3) Pk+1 = (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 Approachby 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: ShowP1: Assume Pk: Prove
Pk+1: Prove Pk+1: Q.E.D. Prove: ShowP1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D. Prove: ShowP1: Assume Pk: Prove Pk+1: Q.E.D.
Prove: ShowP1: Assume Pk: Prove Pk+1: Prove Pk+1: Q.E.D. Worksheet
Exercise # 1 Prove: ShowP1: Assume Pk: Prove Pk+1: Prove Pk+1:
Q.E.D. Worksheet Exercise # 2 Prove: ShowP1: Assume Pk: Prove Pk+1:
Prove Pk+1: Q.E.D. Worksheet Exercise # 3 Prove: ShowP1: Assume Pk:
Prove Pk+1: Prove Pk+1: Q.E.D. Worksheet Exercise # 4 Prove:
ShowP1: Assume Pk: Prove Pk+1: Q.E.D. Prove: ShowP1: Assume Pk:
Prove Pk+1: Worksheet Exercise # 5 Prove: ShowP1: Assume Pk: Prove
Pk+1: Prove Pk+1: Q.E.D. Worksheet Exercise # 6 Prove: ShowP1:
Assume Pk: Prove Pk+1: Prove Pk+1: Q.E.D. Worksheet Exercise # 7
Prove: ShowP1: Assume Pk: Prove Pk+1: Prove Pk+1: Q.E.D. Worksheet
Exercise # 8 Prove: ShowP1: Assume Pk: Prove Pk+1: Prove Pk+1:
Q.E.D. Worksheet Exercise # 9 Prove: ShowP1: 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: ShowP1: Assume Pk: Prove Pk+1: Prove Pk+1: Q.E.D.