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Patterns and Inductive Reasoning
Objectives
Use Logic to Understand Patterns.
Understand and Use Inductive Reasoning.
Form Conjectures and Find Counterexamples.
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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.
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Using Logic
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 , or 24.
3 + 1 = 4
6 + 3 = 9
= 18
4 + 2 = 6
9 + 4 = 13
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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.
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Using Logic
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.
3 × 2 = 6
18 × 2 = 36
108 × 2 = 216
6 × 3 = 18
36 × 3 = 108
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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.
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7. 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.
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8. Using Inductive Reasoning
1, 1, 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 , 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.)
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9. 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.
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10. 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.
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11. 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.
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Forming Conjectures
Study the list of circles. Use the pattern to answer the questions. a. Make a conjecture about the color of the 11th circle. Solution Odd-numbered circles are red; even-numbered circles are blue. Since 11 is an odd number, we conjecture that the 11th circle is red.
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Forming Conjectures
b. Make a conjecture about the number of regions in the 11th 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 11th circle has 11(2) or 22 regions formed by diameters.
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Forming Conjectures
c. Make a conjecture about the appearance of the 11th circle. Solution The appearance of the 11th circle is red with 22 regions formed by diameters.
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Forming Conjectures
d. Make a conjecture about the appearance of the 30th circle. Solution The 30th circle is blue, since 30 is an even number. The 30th circle has 30(2) or 60 regions formed by diameters. The appearance of the 30th circle is a blue circle with 60 regions formed by diameters.
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Definition
A counterexample is an example that shows that a conjecture is false, or incorrect.
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17. 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.
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18. 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.
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