1 Topic 1.4.2 Inductive and Deductive Reasoning Inductive and Deductive Reasoning

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Topic 1.4.2Topic 1.4.2

Inductive and Deductive

Reasoning

Inductive and Deductive

Reasoning

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Topic1.4.2

Inductive and Deductive ReasoningInductive and Deductive Reasoning

California Standards:24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

What it means for you:You’ll identify inductive and deductive reasoning in math problems, and you’ll find how to use counterexamples to disprove a rule.Key words:• inductive reasoning• deductive reasoning• counterexample

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Topic1.4.2


There are different types of mathematical reasoning.

Two types mentioned in the California math standards are inductive reasoning and deductive reasoning.

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Topic1.4.2

Inductive Reasoning Means Finding a General Rule


Inductive reasoning means finding a general rule by considering a few specific cases.

For example, look at this sequence of square numbers: 1, 4, 9, 16, 25, 36...

The difference between the 1st and 2nd terms is 4 – 1 = 3.

The difference between the 2nd and 3rd terms is 9 – 4 = 5.

The difference between the 3rd and 4th terms is 16 – 9 = 7.

The difference between the 4th and 5th terms is 25 – 16 = 9.

If you look at the differences between successive terms, you find this:

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Topic1.4.2

Using inductive reasoning, you might conclude that:


The difference between successive square numbers is always odd, and each difference is 2 greater than the previous one.

Watch out though — this doesn’t actually prove the rule.

This rule does look believable, but to prove it you’d have to use algebra.

If you look at these differences: 3, 5, 7, 9

… there’s a pattern — each difference is an odd number, and each one is 2 greater than the previous difference.

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Topic1.4.2

Guided Practice

Solution follows…


Use inductive reasoning to work out an expression for the nth term (xn) of these sequences. For example, the formula for the nth term of the sequence 1, 2, 3, 4,... is xn = n.

1. 2, 3, 4, 5,... 2. 11, 12, 13, 14,...

3. 2, 4, 6, 8,... 4. –1, –2, –3, –4,...

In Exercises 5-6, predict the next number in each pattern.

5. 1 = 12, 1 + 3 = 22, 1 + 3 + 5 = 32, 1 + 3 + 5 + 7 = ?

6. 1, 1, 2, 3, 5, 8, 13, …

xn = n + 1 xn = n + 10

xn = 2n xn = –n

42

8 + 13 = 21

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Topic1.4.2

One Counterexample Proves That a Rule Doesn’t Work


If you are testing a rule, you only need to find one counterexample (an example that does not work) to prove that the rule is not true.

Once you have found one counterexample, you don’t need to look for any more — one is enough.

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Topic1.4.2

Example 1

Solution follows…


Decide whether the following statement is always true:“2n + 1 is always a prime number, if n is a natural number.”

Solution

At first, the rule looks believable.

If n = 1: 2n + 1 = 21 + 1 = 2 + 1 = 3.This is a prime number, so the rule holds for n = 1.

If n = 2: 2n + 1 = 22 + 1 = 4 + 1 = 5.This is a prime number, so the rule holds for n = 2.

Solution continues…

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Topic1.4.2

Example 1


Decide whether the following statement is always true:“2n + 1 is always a prime number, if n is a natural number.”

Solution (continued)

If n = 3: 2n + 1 = 23 + 1 = 8 + 1 = 9.

This is not a prime number, so the rule doesn’t hold for n = 3.

So a counterexample is n = 3, and this proves that the rule is not always true.

Once you’ve found one counterexample, you don’t need to find any more.

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Topic1.4.2

Guided Practice

Solution follows…


Give a counterexample to disprove each of the following statements.

7. All odd numbers are of the form 4n + 1, where n is a natural number.

8. If a number is divisible by both 6 and 3, then it is divisible by 12.

9. |x + 4| 4

10. The difference between any two square numbers is always odd.

11. The difference between any two prime numbers is always even.

For example, 3.

For example, 6.

For example, x = –4.

For example, 25 – 9 = 16.

For example, 3 – 2 = 1.

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Topic1.4.2

Guided Practice

Solution follows…


Give a counterexample to disprove each of the following statements.

12. The difference between two consecutive cube numbers is always prime.

13. All quadrilaterals are squares.

14. All angles are right angles.

15. All prime numbers are odd.

16. A number is always greater than its multiplicative inverse.

For example, 63 – 53 = 216 – 125 = 91 (= 7 × 13).

For example, rectangles, trapezoids, kites.

For example, any obtuse or acute angle.

For example, 2.

For example, ¼ < 4.

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Topic1.4.2

Deductive Reasoning Means Applying a General Rule


Deductive reasoning is almost the opposite of inductive reasoning.

In deductive reasoning, you use a general rule to find out a specific fact.

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Topic1.4.2

Example 2

Solution follows…


A number is a multiple of 3 if the sum of its digits is a multiple of 3. Use this information to decide whether 96 is a multiple of 3.

Solution

The sum of the digits is 9 + 6 = 15, which is divisible by 3.

The statement says that a number is a multiple of 3 if the sum of its digits is a multiple of 3.

Using deductive reasoning, that means that you can say that 96 is divisible by 3.

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Topic1.4.2

Guided Practice

Solution follows…


Use deductive reasoning to work out the 10th term of these sequences:

17. xn = n 18. xn = 6n

19. xn = 2n – 1 20. xn = 3n – 1

21. xn = 20n + 1 22. xn = n(n + 1)

Use deductive reasoning to reach a conclusion:

23. Ivy is older than Peter. Stephen is younger than Peter.

24. Lily lives in Maryland. Maryland is in the United States.

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19

201

60

29

110

Ivy is older than Stephen.

Lily lives in the United States.

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Topic1.4.2

Independent Practice

Solution follows…


Use inductive reasoning in Exercises 1–2.

1. Give the next three numbers of the sequence: 25, 29, 34, 40, …

2. Write an expression for the nth term (xn) of the sequence:

24, 72, 216, 648, …

47, 55, 64

xn = 3xn–1

or, xn = 24 • 3xn–1

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Solution follows…


3. Audrey needs $650 to buy a digital camera. Her savings account shows the following balances:

If the pattern continues, at the start of which month will she be able to buy the digital camera? Use inductive reasoning.

Date Balance

Jan 1

Feb 1

Mar 1

Apr 1

$100.00

$150.00

$250.00

$400.00

June

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Topic1.4.2


Solution follows…


Use deductive reasoning in exercises 4-5.

4. Find the first five terms of the sequence xn = 3n.

5. Find the first five terms of the sequence xn = n(n – 1).

3, 6, 9, 12, 15

0, 2, 6, 12, 20

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Round UpRound Up


Inductive reasoning means that you can make a general rule without having to check every single value — so it saves you a lot of work.

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1 Topic 1.4.2 Inductive and Deductive Reasoning Inductive and Deductive Reasoning