2-1 Patterns and Inductive Reasoning

Quick Review

You use inductive reasoning when you make conclusions based on patterns you observe. A conjecture is a conclusion you reach using inductive reasoning. A counterexample is an example that shows a conjecture is incorrect.

Example

Describe the pattern. What are the next two terms in the sequence?

1 , − 3 , 9 , − 27 , … 1 comma negative 3 comma 9 comma negative 27 comma dot dot dot

Each term is − 3 negative 3  times the previous term. The next two terms are − 27 × ( − 3 ) = 81 negative 27 times open negative 3 close equals 81  and 81 × ( − 3 ) = − 243 . 81 times open negative 3 close , equals negative 243 , .

Exercises

Find a pattern for each sequence. Describe the pattern and use it to show the next two terms.

8. 1000, 100, 10, …
9. 5 , − 5 , 5 , − 5 , … 5 comma negative 5 comma 5 comma negative 5 comma dot dot dot
10. 34, 27, 20, 13, …
11. 6, 24, 96, 384, …

Find a counterexample to show that each conjecture is false.

12. The product of any integer and 2 is greater than 2.
13. The city of Portland is in Oregon.

2-2 Conditional Statements

Quick Review

A conditional is an if-then statement. The symbolic form of a conditional is p → q , p rightwards arrow q comma  where p is the hypothesis and q is the conclusion.

• To find the converse, switch the hypothesis and conclusion of the conditional ( q → p ) . open q rightwards arrow p close .
• To find the inverse, negate the hypothesis and the conclusion of the conditional ( ˜ p → ˜ q ) . open tilde p rightwards arrow tilde q close .
• To find the contrapositive, negate the hypothesis and the conclusion of the converse ( ˜ q → ˜ p ) . open tilde q rightwards arrow tilde p close .

Example

What is the converse of the conditional statement below? What is its truth value?

If you are a teenager, then you are younger than 20.

Converse: If you are younger than 20, then you are a teenager.

A 7-year-old is not a teenager. The converse is false.

Exercises

Rewrite each sentence as a conditional statement.

14. All motorcyclists wear helmets.
15. Two nonparallel lines intersect in one point.
16. Angles that form a linear pair are supplementary.
17. School is closed on certain holidays.

Write the converse, inverse, and contrapositive of the given conditional. Then determine the truth value of each statement.

18. If an angle is obtuse, then its measure is greater than 90 and less than 180.
19. If a figure is a square, then it has four sides.
20. If you play the tuba, then you play an instrument.
21. If you baby-sit, then you are busy on Saturday night.

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