Patterns and Inductive Reasoning Objectives 1 Use Logic

Patterns and Inductive Reasoning Objectives 1. Use Logic to Understand Patterns. 2. Understand Use Inductive Reasoning. 3. Form Conjectures and Find Counterexamples. Copyright © 2014 Pearson Education, Inc. Slide 1

Example Using Logic Look for a pattern in each list. Then use this pattern to predict the next number. a. 3, 4, 6, 9, 13, 18, _______ b. 3, 6, 18, 36, 108, 216, _______ Solution Because 3, 4, 6, 9, 13, 18, ___ is increasing relatively slowly, let’s try addition as a possible basis for this pattern. Copyright © 2014 Pearson Education, Inc. Slide 2

Using Logic Example 3 4 6 3+1=4 9 6+3=9 4+2=6 13 18 13 + 5 = 18 9 + 4 = 13 From these observations, we conclude that each number after the first is obtained by adding a counting number to the previous number. The additions begin with 1 and continue through each successive counting number. Using this pattern, the next number is 18 + 6, or 24. Copyright © 2014 Pearson Education, Inc. Slide 3

Example Using Logic b. 3, 6, 18, 36, 108, 216, _______ Solution Because 3, 6, 18, 36, 108, 216, _______ is increasing relatively rapidly, let’s try multiplication as a possible basis for this pattern. Copyright © 2014 Pearson Education, Inc. Slide 4

Example 3 6 3× 2=6 Using Logic 18 36 108 216 108 × 2 = 216 18 × 2 = 36 6 × 3 = 18 36 × 3 = 108 From these observations, we conclude that each number after the first is obtained by multiplying the previous number by 2 or by 3. The multiplications begin with 2 and then alternate, multiplying by 2, then 3, then 2, then 3, and so on. Using this pattern, the next number is 216 × 3, or 648. Copyright © 2014 Pearson Education, Inc. Slide 5

Definition Inductive reasoning is the process of arriving at a general conclusion based on observing patterns or observing specific examples. Although inductive reasoning is a powerful method of drawing conclusions, we can never be absolutely certain that these conclusions are true. For this reason, the conclusions are called conjectures or educated guesses. Copyright © 2014 Pearson Education, Inc. Slide 6

Example Using Inductive Reasoning Look for a pattern in each list. Then use this pattern to predict the next number. a. 1, 1, 2, 3, 5, 8, 13, 21, _______ b. 23, 54, 95, 146, 117, 98, _______ Solution Starting with the third number in the list, let’s form our observations by comparing each number with the two numbers that immediately precede it. Copyright © 2014 Pearson Education, Inc. Slide 7

Example 1, Using Inductive Reasoning 2, 3, 5, 8, 13, 21 Notice that the first two numbers are 1. Each number thereafter is the sum of the two preceding numbers. Using this pattern, the next number is 13 + 21, or 34. (The numbers 1, 1, 2, 3, 5, 8, 13, 21, and 34 are the first nine terms of a list of numbers called the Fibonacci sequence. ) Copyright © 2014 Pearson Education, Inc. Slide 8

Example Using Inductive Reasoning b. 23, 54, 95, 146, 117, 98, _______ Notice that starting with the second number, we obtain the first digit of the number by adding the digits of the previous number. We obtain the last digit of the number by adding 1 to the final digit of the preceding number. Applying this pattern to the number that follows 98, the first part of the number is 9 + 8, or 17. The last digit is 8 + 1, or 9. Thus, the next number in the list is 179. Copyright © 2014 Pearson Education, Inc. Slide 9

Example Using Inductive Reasoning Notice two patterns in this sequence of figures. Use the patterns to draw the next figure in the sequence. Solution The figures alternate between circles and squares. The next figure will be a circle. Copyright © 2014 Pearson Education, Inc. Slide 10

Example Using Inductive Reasoning The second pattern in the four regions is the dot pattern. This dot pattern remains the same except that it rotates counterclockwise as we follow the figures from left to right. The next figure should be a circle with a single dot in the right-hand region, two dots in a bottom region, three dots in the left-hand region, and no dots in the top region. Copyright © 2014 Pearson Education, Inc. Slide 11

Example Forming Conjectures Study the list of circles. Use the pattern to answer the questions. a. Make a conjecture about the color of the 11 th circle. Solution Odd-numbered circles are red; even-numbered circles are blue. Since 11 is an odd number, we conjecture that the 11 th circle is red. Copyright © 2014 Pearson Education, Inc. Slide 12

Example Forming Conjectures b. Make a conjecture about the number of regions in the 11 th circle. Solution If we number the circles in the list, each numbered circle has twice as many regions as its number in the list. Thus, the 11 th circle has 11(2) or 22 regions formed by diameters. Copyright © 2014 Pearson Education, Inc. Slide 13

Example Forming Conjectures c. Make a conjecture about the appearance of the 11 th circle. Solution The appearance of the 11 th circle is red with 22 regions formed by diameters. Copyright © 2014 Pearson Education, Inc. Slide 14

Example Forming Conjectures d. Make a conjecture about the appearance of the 30 th circle. Solution The 30 th circle is blue, since 30 is an even number. The 30 th circle has 30(2) or 60 regions formed by diameters. The appearance of the 30 th circle is a blue circle with 60 regions formed by diameters. Copyright © 2014 Pearson Education, Inc. Slide 15

Definition A counterexample is an example that shows that a conjecture is false, or

Example Finding a Counterexample Find a counterexample to show that each conjecture is false. a. Conjecture: The product of two numbers is always greater than either number. Solution Sometimes this conjecture is true. 3(4) = 12 For the product 12, it is true that 12 > 4 and 12 > 3. Counterexample: – 2(4) = – 8. Notice that The conjecture is false because it is not always true. Copyright © 2014 Pearson Education, Inc. Slide 17

Example Finding a Counterexample Find a counterexample to show that each conjecture is false. b. Conjecture: All apples are red. Solution Simply find one apple that is not red. For example, Granny Smith apples are green. Copyright © 2014 Pearson Education, Inc. Slide 18