1.3.1B Inductive Reasoning
Transcript of 1.3.1B Inductive Reasoning
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Inductive Reasoning
Objectives:
The student is able to (I can):
Use inductive reasoning to identify patterns and make conjectures
Find counterexamples to disprove conjectures
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Find the next item in the sequence:
1. December, November, October, ...
SeptemberSeptemberSeptemberSeptember
2. 3, 6, 9, 12, ...
15151515
3. , , , ...
4. 1, 1, 2, 3, 5, 8, ...
13 13 13 13 This is called the Fibonacci This is called the Fibonacci This is called the Fibonacci This is called the Fibonacci sequence.sequence.sequence.sequence.
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inductive reasoning
conjecture
Reasoning that a rule or statement is true because specific cases are true.
A statement believed true based on inductive reasoning.
Complete the conjecture:
The product of an odd and an even number is ______ .
To do this, we consider some examples:
(2)(3) = 6 (4)(7) = 28 (2)(5) = 10
eveneveneveneven
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counterexample
If a conjecture is true, it must be true for every case. Just one exampleJust one exampleJust one exampleJust one example for which the conjecture is false will disprove it.
A case that proves a conjecture false.
Example: Find a counterexample to the conjecture that all students who take Geometry are 10th graders.
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Examples
To Use Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a counterexample to disprove it.
Show that each conjecture is false by giving a counterexample.
1. The product of any two numbers is greater than the numbers themselves.
((((----1)(5) = 1)(5) = 1)(5) = 1)(5) = ----5555
2. Two complementary angles are not congruent.
45 and 4545 and 4545 and 4545 and 45
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Sometimes we can use inductive reasoning to solve a problem that does not appear to have a pattern.
Example: Find the sum of the first 20 odd numbers.
Sum of first 20 odd numbers?
1
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7
1
4
9
16
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Sometimes we can use inductive reasoning to solve a problem that does not appear to have a pattern.
Example: Find the sum of the first 20 odd numbers.
Sum of first 20 odd numbers?
1
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7
1
4
9
16
12
22
32
42
202 = 400
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These patterns can be expanded to find the nth term using algebra. When you complete these sequences by applying a rule, it is called a functionfunctionfunctionfunction.
Examples: Find the missing terms and the rule.
To find the pattern when the difference between each term is the same, the coefficient of n is the difference between each term, and the value at 0 is what is added or subtracted.
1 2 3 4 5 8 20 n
-3 -2 -1 0 1 4 16 n 4
1 2 3 4 5 8 20 n
32 39 46 53 60 81 165 7n+25