Inductive Reasoning

Objectives:

The student is able to (I can):

Use inductive reasoning to identify patterns and make conjectures

Find counterexamples to disprove conjectures

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.

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

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.

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

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

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

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