2-1 Conjectures and Counterexamples Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. 1. Costs: $4.50, $6.75, $9.00 . 3. SOLUTION: SOLUTION: $6.75 = $4.50 + $2.25 In each figure, the shading moves to the next point $9 = $6.75 + $2.25 clockwise. The next figure in the sequence is: Each cost is $2.25 more than the previous cost. $9.00 + $2.25 = $11.25 The next item in the sequence is $11.25. ANSWER: ANSWER: Each cost is $2.25 more than the previous cost; In each figure, the shading moves to the next point $11.25. clockwise. 2. Appointment times: 10:15 a.m., 11:00 a.m., 11:45 a.m., . SOLUTION: 11:00 a.m. = 10:15 a.m. + 0:45 11:45 a.m. =11:00 a.m. + 0:45 Each time is 45 minutes later than the previous time. Then, 11:45 a.m. + 0:45 = 12:30 p.m. The next item in the sequence is 12:30 p.m. ANSWER: Each time is 45 minutes later than the previous time; 12:30 P.M. eSolutions Manual - Powered by Cognero Page 1 2-1 Conjectures and Counterexamples 6. 2, 6, 14, 30, 62, . SOLUTION: 4. Consider the difference in the terms. 6 – 2 = 4 SOLUTION: 14 – 6 = 8 Each figure in the pattern has an additional circle 30 – 14 = 16 around the outside. The first figure is 1 circle, then 2 62 – 30 = 32 circles, and then 3 circles. The next figure in the sequence is: Since the difference are not the same, the sequence is not arithmetic. However, each line, the answer is a factor of two more than the previous one. 6 = 2(2) + x . 14 = 2(6) + x It has 4 circles. 30 = 2(14) + x ANSWER: 62 = 2(30) + x Each figure in the pattern has an additional circle Find the ? that make each statement true? around the outside. 5. 3, 3, 6, 9, 15, . Then, 6 = 2 + 2(2) SOLUTION: 14 = 2 + 2(6) 6 = 3 + 3 30 = 2 + 2(14) 9 = 3 + 6 62 = 2 + 2(30). 15 = 6 + 9 Each element is two more than two times the Each element in the pattern is sum of the previous previous element. two elements. 2 + 2(62) = 126 9 + 15 = 24 The next item in the sequence is 126. The next item in the sequence is 24. ANSWER: ANSWER: Each element is two more than two times the Each element in the pattern is the sum of the previous previous element; 126. two elements; 24. eSolutions Manual - Powered by Cognero Page 2 2-1 Conjectures and Counterexamples Make a conjecture about each value or 10. the relationship between and if M is the geometric relationship. midpoint of and P is the midpoint of 7. the product of two even numbers SOLUTION: SOLUTION: Given: M is the midpoint of and P is the midpoint The product of two even numbers is an even number. of Examples: 2 × 2 = 4 2 × 6 = 12 2 × 8 = 16 … ANSWER: The product of two even numbers is an even number. Let the distance from A to P be x. Then the distance from A to M is 2x and from M to B is 2x. 8. the relationship between a and b if a + b = 0 The midpoint of M and B would be x away from either endpoint. SOLUTION: So, is three times as long as . If a + b =0, then a and b are additive inverses. ANSWER: Examples: is three times as long as . 3 + (–3)= 0, 3 and –3 are additive inverses 4 + (–4) = 0, 4 and –4 are additive inverses 11. SMARTPHONES Refer to the table of the number 1000 + (–1000) = 0, 1000 and –1000 are additive of smartphone users in the United States by year. inverses… a. Make a graph that shows U.S. smartphone use from 2010–2015. ANSWER: b. Make a conjecture about U.S. smartphone use in a and b are additive inverses. 2018. 9. the relationship between the set of points in a plane equidistant from point A SOLUTION: The set of points in a plane equidistant from point A is a circle. SOLUTION: a. Use a scatter plot to display the data. Label the ANSWER: horizontal axis "Year" and the vertical axis "Users The set of points in a plane equidistant from point A is (millions)." Plot the data on a coordinate grid. a circle. eSolutions Manual - Powered by Cognero Page 3 2-1 Conjectures and Counterexamples b. Sample answer: About 272,000,000 Americans b. Look for patterns in the data. The number of will use smartphones in 2018. smartphone users increases each year. Since the number of users appears to increase linearly, draw a CRITIQUE ARGUMENTS Find a counterexample to show that each conjecture is line of best fit that closely models the data. Choose false. two points on the line of best fit to determine the 12. If and are complementary angles, then they equation for the line. share a common side. Sample answer: SOLUTION: Draw two angles that are not connected, but still sum to 90 degrees. ANSWER: Use two points on the line to find the slope. Substitute 2018 for x in the equation to find the number of smart phones in millions. So, in 2018 about 272,000,000 Americans will have smart phones. ANSWER: a. eSolutions Manual - Powered by Cognero Page 4 2-1 Conjectures and Counterexamples 13. If a ray intersects a segment at its midpoint, then the 15. 3, 6, 9, 12, 15 ray is perpendicular to the segment. SOLUTION: SOLUTION: 6 = 3 + 3 Draw a ray that intersects a segment, but not at a 90 9 = 3 + 6 degree angle. 12 = 3 +9 15 = 3 +12 Each element in the pattern is three more than previous element. 3 + 15 = 18 ANSWER: The next element in the sequence is 18. ANSWER: Each element in the pattern is three more than the previous element; 18. Write a conjecture that describes the pattern in each sequence. Then use your conjecture to 16. 4, 8, 12, 16, 20 find the next item in the sequence. 14. 0, 2, 4, 6, 8 SOLUTION: 8 = 4 + 4 SOLUTION: 12 = 4 + 8 2 = 0 + 2 16 = 4 + 12 4 = 2 + 2 20 = 4 +16 6 = 2 + 4 Each element in the pattern is four more than the 8 = 2 + 6 previous element. Each element in the pattern is two more than the 4 + 20 =24 previous element. The next element in the sequence is 24. 2 + 8 =10 The next element in the sequence is 10. ANSWER: Each element in the pattern is four more than the ANSWER: previous element; 24. Each element in the pattern is two more than the previous element; 10. 17. 2, 22, 222, 2222 SOLUTION: Each element has an additional two as part of the number. 22 = 2 + 2·10 222 = 2 + 22·10 2222 = 2 + 222·10 Thus the next term is 2 + 2222·10 or 22222. ANSWER: Each element has an additional two as part of the number; 22222. eSolutions Manual - Powered by Cognero Page 5 2-1 Conjectures and Counterexamples 18. 1, 4, 9, 16 20. Arrival times: 3:00 P.M., 12:30 P.M., 10:00 A.M., . SOLUTION: SOLUTION: 1 = 12 12:30 P.M.= 3:00 P.M., – 2:30 4 = 22 10:00 A.M.= 12:30 P.M. – 2:30 9 = 32 Each arrival time is 2 hours and 30 minutes prior to 16 = 42 the previous arrival time. Each element is the square of increasing natural numbers. 10:00 A.M. – 2:30 = 7:30 A.M. 2 5 = 25 The next element in the sequence is 25. The next arrival time in the sequence is 7:30 A.M. ANSWER: ANSWER: Each element is the square of increasing natural Each arrival time is 2 hours and 30 minutes prior to numbers; 25. the previous arrival time; 7:30 A.M. 19. 21. Percent humidity: 100%, 93%, 86%, . SOLUTION: SOLUTION: 93% = 100% – 7% 86% = 93% – 7% Each percentage is 7% less than the previous percentage. 86% – 7% = 79% Each element is one half the previous element. The next percentage in the sequence is 79%. ANSWER: Each percentage is 7% less than the previous The next element in the sequence is . percentage; 79%. ANSWER: Each element is one half the previous element; . eSolutions Manual - Powered by Cognero Page 6 2-1 Conjectures and Counterexamples 22. Work-out days: Sunday, Tuesday, Thursday, . SOLUTION: Sample answer: 25. Tuesday is two days after Sunday. SOLUTION: Thursday is two days after Tuesday. In each figure, the shading moves to the next area of Each work out day is two days after the previous the figure counter clockwise. The next figure in the day. sequence is So, the next work out day in the sequence is Saturday. ANSWER: . Sample answer: each work out day is two days after ANSWER: the previous day; Saturday. In each figure, the shading moves to the next area of 23. Club meetings: January, March, May, . the figure counter clockwise. SOLUTION: March is two month after January and May is two months after March. Each meeting is two months after the previous meeting. The next month is in the sequence is July. ANSWER: 26. Each meeting is two months after the previous meeting; July. SOLUTION: The first polygon has 3 sides, then 4 sides, and then 5 sides.Each figure in the pattern is the next largest regular polygon.