6.2 Arithmetic Sequences

Transcription

1 6.2 Arithmetic Sequences A sequence like 2, 5, 8, 11,, where the difference between consecutive terms is a constant, is called an arithmetic sequence. In an arithmetic sequence, the first term, t 1, is denoted by the letter a. Each term after the first is found by adding a constant, called the common difference, d, to the preceding term. INVESTIGATE & INQUIRE For about 200 years, the Gatineau River was used as a highway by logging companies. Logs from the Canadian Shield were floated down the river to the Ottawa River. It has been estimated that 2% of the hundreds of millions of logs that floated down the Gatineau sank. Those that sank below the oxygen level are perfectly preserved and are now being harvested by water loggers, who wear scuba-diving gear. The pressure that a diver experiences is the sum of the pressure of the atmosphere and the pressure of the water. The increase in pressure with depth under water follows an arithmetic sequence. If a diver enters the water when the atmospheric pressure is 100 kpa (kilopascals), the pressure at a depth of 1 m is about 110 kpa. At a depth of 2 m, the pressure is about 120 kpa, and so on. 1. Copy and complete the table for this sequence. Term Pressure (kpa) Pressure (kpa) Expressed Using 100 and 10 Pressure Expressed Using a and d t 1 t 2 t 3 t 4 t (10) a a + 2. What are the values of a and d for this sequence? 3. When you write an expression for a term using the letters a and d, you are writing a formula for the term. What is the formula for t 6? t 8? t 9? 436 MHR Chapter 6

2 4. Evaluate t 8 and t The Gatineau River has maximum depth of 35 m. What pressure would a diver experience at this depth? EXAMPLE 1 Writing Terms of a Sequence Given the formula for the nth term of an arithmetic sequence, = 2n + 1, write the first 6 terms. SOLUTION 1 Paper-and-Pencil Method = 2n + 1 t 1 = 2(1) + 1 = 3 t 2 = 2(2) + 1 = 5 t 3 = 2(3) + 1 = 7 t 4 = 2(4) + 1 = 9 t 5 = 2(5) + 1 = 11 t 6 = 2(6) + 1 = 13 The first 6 terms are 3, 5, 7, 9, 11, and 13. SOLUTION 2 Graphing-Calculator Method Use the mode settings to choose the Seq (sequence) graphing mode. Use the sequence function from the LIST OPS menu to generate the first 6 terms. The first 6 terms are 3, 5, 7, 9, 11, and 13. Note that the arithmetic sequence defined by = 2n + 1, or f(n) = 2n + 1, in Example 1, is a linear function, as shown by the following graphs n 6.2 Arithmetic Sequences MHR 437

3 EXAMPLE 2 Determining the Value of a Term Given the formula for the nth term, find t 10. a) = 7 + 4n b) f(n) = 5n 8 SOLUTION 1 Pencil-and-Paper Method a) = 7 + 4n b) f(n) = 5n 8 t 10 = 7 + 4(10) f(10) = 5(10) 8 = = 50 8 = 47 = 42 SOLUTION 2 Graphing-Calculator Method Use the mode settings to choose the Seq (sequence) graphing mode. Use the sequence function from the LIST OPS menu to generate the 10th term. a) b) Note that the general arithmetic sequence is a, a + d, a + 2d, a + 3d, where a is the first term and d is the common difference. t 1 = a t 2 = a + d t 3 = a + 2d. = a + (n 1)d, where n is a natural number. Note that d is the difference between any successive pair of terms. For example, t 2 t 1 = (a + d ) a = d t 3 t 2 = (a + 2d ) (a + d ) = a + 2d a d = d 438 MHR Chapter 6

4 EXAMPLE 3 Finding the Formula for the nth Term Find the formula for the nth term,, and find t 19, for the arithmetic sequence 8, 12, 16, SOLUTION For the given sequence, a = 8 and d = 4. = a +(n 1)d Substitute known values: = 8 + (n 1)4 Expand: = 8 + 4n 4 Simplify: = 4n + 4 Three ways to find t 19 are as follows. Method 1 Method 2 Method 3 = a +(n 1)d = 4n + 4 Use a graphing calculator. t 19 = a + (19 1)d t 19 = 4(19) + 4 = a + 18d = = (4) = 80 = = 80 So, = 4n + 4 and t 19 = 80. EXAMPLE 4 Finding the Number of Terms How many terms are there in the following sequence? 3, 2, 7,, 152 SOLUTION For the given sequence, a = 3, d = 5, and = 152. Substitute the known values in the formula for the general term and solve for n. 6.2 Arithmetic Sequences MHR 439

5 = a + (n 1)d Substitute known values: 152 = 3 + (n 1)5 Expand: 152 = 3 + 5n 5 Simplify: 152 = 5n 8 Solve for n: = 5n = 5n = 5n 5 The sequence has 32 terms. 32 = n EXAMPLE 5 Finding Given Two Terms In an arithmetic sequence, t 7 = 121 and t 15 = 193. Find the first three terms of the sequence and. SOLUTION Substitute known values in the formula for the nth term to write a system of equations. Then, solve the system. = a + (n 1)d Write an equation for t 7 : 121 = a + (7 1)d 121 = a + 6d (1) Write an equation for t 15 : 193 = a + (15 1)d 193 = a + 14d (2) Subtract (1) from (2): 72 = 8d Solve for d: 9 = d Substitute 9 for d in (1): 121 = a + 6(9) Solve for a: 121 = a = a Since a = 67 and d = 9, the first three terms of the sequence are 67, 76, and 85. To find, substitute 67 for a and 9 for d in the formula for the nth term. = a + (n 1)d = 67 + (n 1)9 Simplify: = n 9 = 9n + 58 So, the first three terms are 67, 76, and 85, and = 9n MHR Chapter 6 The (1) shows that we are naming the equation as equation one. You can check by substituting 67 for a and 9 for d in (2).

6 Key Concepts The general arithmetic sequence is a, a + d, a + 2d, a + 3d,, where a is the first term and d is the common difference. The formula for the nth term, or f(n), of an arithmetic sequence is = a + (n 1)d, where n is a natural number. Communicate Your Understanding 1. Given the formula for the nth term of an arithmetic sequence, = 4n 3, describe how you would find the first 6 terms. 2. a) Describe how you find the formula for the nth term of the arithmetic sequence 3, 8, 13, 18, b) Describe how you would find t 46 for this sequence. 3. Describe how you would find the number of terms in the sequence 5, 10, 15,, Given that t 5 = 11 and t 12 = 25 for an arithmetic sequence, describe how you would find for the sequence. Practise A 1. Find the next three terms of each arithmetic sequence. a) 3, 7, 11, b) 33, 27, 21, c) 23, 18, 13, d) 25, 18, 11, e) 5.8, 7.2, 8.6, f),,, Given the formula for the nth term of an arithmetic sequence, write the first 4 terms. a) = 3n + 5 b) f(n) = 2n 7 c) = 4n 1 d) f(n) = 6 n n + 3 e) = 5n 2 f) f(n) = 2 3. Given the formula for the nth term of an arithmetic sequence, write the indicated term. a) = 2n 5; t 11 b) = 4 + 3n; t 15 c) f(n) = 4n + 5; t 10 d) f(n) = 0.1n 1; t 25 2n 1 e) = 2.5n + 3.5; t 30 f) f(n) = ; t Determine which of the following sequences are arithmetic. If a sequence is arithmetic, write the values of a and d. a) 5, 9, 13, 17, b) 1, 6, 10, 15, 19, c) 2, 4, 8, 16, 32, d) 1, 4, 7, 10, e) 1, 1, 1, 1, 1, f),,,, g) 4, 2.5, 1, 0.5, h) y, y 2, y 3, y 4, i) x, 2x, 3x, 4x, j) c, c + 2d, c + 4d, c + 6d, 6.2 Arithmetic Sequences MHR 441

7 5. Given the values of a and d, write the first five terms of each arithmetic sequence. a) a = 7, d = 2 b) a = 3, d = 4 c) a = 4, d = 6 d) a = 2, d = 3 e) a = 5, d = f) a =, d = 2 2 g) a = 0, d = 0.25 h) a = 8, d = x i) t 1 = 6, d = y + 1 j) t 1 = 3m, d = 1 m 6. Find the formula for the nth term and find the indicated terms for each arithmetic sequence. a) 6, 8, 10, ; t 10 and t 34 b) 12, 16, 20, ; t 18 and t 41 c) 9, 16, 23, ; t 9 and t 100 d) 10, 7, 4, ; t 11 and t 22 e) 4, 9, 14, ; t 18 and t f),,, ; t 12 and t g) 5, 1, 7, ; t 8 and t 14 h) 7, 10, 13, ; t 15 and t 30 i) 10, 8, 6, ; t 13 and t 22 j) x, x + 4, x + 8, ; t 14 and t Find the number of terms in each of the following arithmetic sequences. a) 10, 15, 20,, 250 b) 1, 4, 7,, 121 c) 40, 38, 36,, 30 d) 11, 7, 3,, 153 e) 2, 8, 14,, 206 f) 6, 7, 1,, g) x + 2, x + 9, x + 16,, x Find a and d, and then write the formula for the nth term,, of arithmetic sequences with the following terms. a) t 5 = 16 and t 8 = 25 b) t 12 = 52 and t 22 = 102 c) t 50 = 140 and t 70 = 180 d) t 2 = 12 and t 5 = 9 e) t 7 = 37 and t 10 = 121 f) t 8 = 166 and t 12 = 130 g) t 4 = 2.5 and t 15 = 6.9 h) t 3 = 4 and t 21 = 5 9. The third term of an arithmetic sequence is 24 and the ninth term is 54. a) What is the first term? b) What is the formula for the nth term? 10. The fourth term of an arithmetic sequence is 14 and the eleventh term is 35. a) What are the first four terms? b) What is the formula for the nth term? Apply, Solve, Communicate 11. The graph of an arithmetic sequence is shown. a) What are the first five terms of the sequence? b) What is t 50? t 200? n 442 MHR Chapter 6

8 B 12. Find the common difference of the sequence whose formula for the nth term is = 2n Copy and complete each arithmetic sequence. Graph versus n for each sequence. a),, 14,, 26 b), 3,,, Olympic Games The modern summer Olympic Games were first held in Athens, Greece, in The games were to be held every four years, so the years of the games form an arithmetic sequence. a) What are the values of a and d for this sequence? b) Research In what years were the games cancelled and why? c) What are the term numbers for the years the games were cancelled? d) What is the term number for the next summer games? 15. Multiples How many multiples of 5 are there from 15 to 450, inclusive? 16. The 18th term of an arithmetic sequence is 262. The common difference is 15. What is the first term of the sequence? 17. Driving Barrie is 60 km north of Toronto by road. If you drive north from Barrie at 80 km/h, how far are you from Toronto by road after a) 1 h? b) 2 h? c) t hours? 18. Inquiry/Problem Solving Comets approach the Earth at regular intervals. For example, Halley s Comet reaches its closest point to the Earth about every 76 years. Comet Finlay is expected to reach its closest point to the Earth in 2009, 2037, and three times between these years. In which years between 2009 and 2037 will Comet Finlay reach its closest point to the Earth? 19. Electrician Amber works as an electrician. She charges $60 for each service call, plus an hourly rate. If she charges $420 for an 8-h service call a) what is her hourly rate? b) how much would she charge for a 5-h service call? 20. Salary Franco is the manager of a health club. He earns a salary of $ a year, plus $200 for every membership he sells. What will he earn in a year if he sells 71 memberships? 88 memberships? 104 memberships? 6.2 Arithmetic Sequences MHR 443

9 21. Ring sizes A ring size indicates a standardized inside diameter of a ring. The table gives the inside diameters for 5 ring sizes. a) Determine the formula for the nth term of the sequence of inside diameters. b) Use the formula to find the inside diameter of a size 13 ring. 22. Displaying merchandise Boxes are stacked in a store display in the shape of a triangle. The numbers of boxes in the rows form an arithmetic sequence. There are 41 boxes in the 3rd row from the bottom. There are 23 boxes in the 12th row from the bottom. a) How many boxes are there in the first (bottom) row? b) What is the formula for the nth term of the sequence? c) What is the maximum possible number of rows of boxes? 23. Application On the first day of practice, the soccer team ran eight 40-m wind sprints. On each day after the first, the number of wind sprints was increased by two from the day before. a) What are the values of a and d for this sequence? b) Write the formula for the nth term of the sequence. c) How many wind sprints did the team run on the 15th day of practice? How many metres was this? 24. Pattern How many dots are in the 51st figure? n = Pattern The U-shapes are made from asterisks. a) How many asterisks are in the 4th diagram? b) What is the formula for the nth term of the sequence in the numbers of asterisks? c) How many asterisks are in the 25th diagram? d) Which diagram contains 139 asterisks? Ring Size 26. Astronomy The time from one full moon to the next is days. The first full moon of a year occurred days into the year. a) How many days into the year did the 9th full moon occur? b) At what time of day did the 9th full moon occur? Inside Diameter (mm) MHR Chapter 6

10 27. The eighth term of an arithmetic sequence is 5.3 and the fourteenth term is 8.3. What is the fifth term? 28. Communication a) Use finite differences to explain why the graph of versus n for an arithmetic sequence is linear. b) Explain why the points on a graph of versus n for an arithmetic sequence are not joined by a straight line. 29. Motion of a pendulum The period of a pendulum is the time it takes to complete one back-and-forth swing. On the Earth, the period, T seconds, is approximately given by the formula T = 2 l, where l metres is the length of the pendulum. If a 1-m pendulum completes its first period at a time of 10:15:30, or 15 min 30 s after 10:00, a) at what time would it complete 100 periods? 151 periods? b) how many periods would it have completed by 10:30:00? 30. Motion of a pendulum Repeat question 29 for a 9-m pendulum on the Earth. 31. Motion of a pendulum The period of a pendulum depends on the acceleration due to gravity, so the period would be different on the moon than on the Earth. On the moon, the period, T seconds, would be given approximately by the formula T = 5 l, where l metres is the length of the pendulum. Repeat question 29 for a 1-m pendulum on the moon. C 32. Measurement The side lengths in a right triangle form an arithmetic sequence with a common difference of 2. What are the side lengths? 33. The sum of the first two terms of an arithmetic sequence is 16. The sum of the second and third terms is 28. What are the first three terms of the sequence? 34. a) How does the sum of the first and fourth terms of an arithmetic sequence compare with the sum of the second and third terms? Explain. b) Find two other pairs of terms whose sums compare in the same way as the two pairs of terms in part a). 35. The first four terms of an arithmetic sequence are 4, 13, 22, and 31. Which of the following is a term of the sequence? Arithmetic Sequences MHR 445

11 36. Algebra The first term of an arithmetic sequence is represented by 3x + 2y. The third term is represented by 7x. Write the expression that represents the second term. 37. Algebra Determine the value of x that makes each sequence arithmetic. a) 2, 8, 14, 4x, b) 1, 3, 5, 2x 1, c) x 2, x + 2, 5, 9, d) x 4, 6, x, e) x + 8, 2x + 8, x, 38. Algebra Find the value of x so that the three given terms are consecutive terms of an arithmetic sequence. a) 2x 1, 4x, and 5x + 3 b) x, 0.5x + 7, and 3x 1 c) 2x, 3x + 1, and x Algebra Find a, d, and for the arithmetic sequence with the terms t 7 = 3 + 5x and t 11 = x. 40. Algebra Show that 1 = d for any arithmetic sequence. A CHIEVEMENT Check Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application 3, 14, 25, and 2, 9, 16, are two arithmetic sequences. Find the first ten terms common to both sequences. LOGIC Power A box contains 5 coloured cubes and an empty space the size of a cube. Use moves like those in checkers. In one move, one cube can slide to an empty space or jump over one cube to an empty space. Find the smallesumber of moves needed to reverse the order of the cubes. 446 MHR Chapter 6