Problem How do you recognize and apply arithmetic sequences and series?

Transcription

1 Arithmetic Series.06-1 Arithmetic Series Topic An arithmetic sequence is a list of numbers in which the difference between two successive terms is a constant. An arithmetic series is the sum of terms in an arithmetic sequence. Problem How do you recognize and apply arithmetic sequences and series? Time 30 to 45 minutes National Math Teaching Standards Number and operations: developing fluency in operations with real numbers Algebra: generalizing patterns using explicitly defined and recursively defined functions Algebra: identifying essential quantitative relationships in a situation and determining the class or classes of functions that might model the relationships Algebra: drawing reasonable conclusions about a situation being modeled Data analysis and probability: being able to determine regression equations using technology tools Materials calculator (TI-8 or TI-83 or the equivalent) INTRODUCTION An arithmetic sequence, written a 1, a, a 3, p a n, a n 1, p is recursively defined by beginning with a real number for a 1 (the first term) and using the formula a n 1 d a n where the constant difference d can be any real Facts On File, Inc. Published by Facts On File, Inc. All electronic storage, reproduction, or transmittal is copyright protected by the publisher.

2 .06- Arithmetic Series MATH ON FILE : Algebra number. If d is positive, the sequence is increasing, if d is negative, the sequence is decreasing. A finite arithmetic series is a sum of the form n a 1 a a 3 p a n a a i. i 1 PROCEDURE AND ANALYSIS Q1. A child nicknamed the Cookie Monster is at a large family reunion. She is delighted to see that everyone brought trays of cookies. She has decided to sample them all. From the first tray, she eats three cookies. These taste so good that from the next tray she eats two more than she did from the first tray. Then, from the third tray, she just cannot resist eating two more than from the second tray. a. How many cookies did she eat from the first three trays? b. If she continues eating like this, how many will she have eaten after she has sampled five trays? After n trays? Q. At the next reunion, the Cookie Monster again finds herself irresistibly drawn to the dessert table, where, once again, she finds many large trays of cookies. Last time, she got in trouble because she ate so many cookies, so she starts with only one from the first tray. But it is so good, she has three more cookies from the second tray than the first, and then, losing control completely, she has three more cookies from the third tray than from the second. a. How many cookies did she eat from the first three trays this time? b. If she continues eating like this, how many will she have eaten after she has sampled five trays? After n trays? Q3. Make up your own Cookie Monster problems and compare your results. She samples a cookies from the first tray. From the second tray, she samples a + d cookies. From each succeeding tray, she samples d more cookies than the preceding tray. a. How many cookies does she sample from the first tray? From the second tray? From the third tray? How many does she take from the nth tray? Facts On File, Inc. Published by Facts On File, Inc. All electronic storage, reproduction, or transmittal is copyright protected by the publisher.

3 MATH ON FILE : Algebra Arithmetic Series.06-3 b. How many cookies has she eaten all together when she has sampled cookies from one tray? From two trays? From three trays? From n trays? CONNECTIONS Since the explicit formula for an arithmetic series is quadratic (see Typical Answers in the Appendix for details), the students can use a calculator s quadratic regression package to find the formulas, given the first few values. Other units involve showing examples of convergent sequences. The triangle numbers are another example of arithmetic sequences. Click here to see typical answers. Facts On File, Inc. Published by Facts On File, Inc. All electronic storage, reproduction, or transmittal is copyright protected by the publisher.

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5 .06. ARITHMETIC SERIES Q1. a. From the first tray, she eats three cookies. From the second, 3 5 cookies, and from the third, 3 7 cookies. So, from the first three trays, she eats cookies. b. From the first five trays she will eat cookies. From the n trays, she will eat 3 (3 ) (3 # ) (3 3 # ) p (3 (n 1) # ) cookies. We need a more usable way to write this sum. Note that: 3 (3 ) (3 # ) (3 3 # ) p (3 (n 1) # ) 3n (0 1 # # 3 # p (n 1) # ) 3n (1 3 p (n 1)) To find a good way to write the last sum, S 1 3 p (n 1), we first use the following scheme to calculate twice the sum, S. S 1 p (n ) (n 1) S (n 1) (n ) p 3 1 S n n p n n (n 1)n This means that S 1 3 p n(n 1) (n 1). So, after she has sampled n trays, she has eaten a total of (n 1)n 3n n n n(n ) cookies. (Alternative answer) A shorter way to solve this particular problem is to try to fit a pattern to the following table. Some students will see this pattern intuitively. after tray number p n the total cookies eaten is p n(n ) Q. a. From the first tray she eats one cookie. From the second tray she eats cookies, and from the third, cookies. So, from the first three trays, she eats cookies. Q. a. From the first tray she eats one cookie. From the second tray she eats cookies, and from the third, cookies. So, from the first three trays, she eats cookies.

6 b. From the first five trays she will eat cookies. In summary, After she has eaten cookies from n trays, she will have eaten a total of 1 (1 3) (1 # 3) (1 3 # 3) p (1 (n 1) # 3) cookies. As above, 1 (1 3) (1 # 3) (1 3 # 3) p (1 (n 1) # 3) 1 # n (0 1 # 3 # 3 3 # 3 p (n 1) # 3) n 3 # (1 3 p (n 1)) n 3 # (n 1)n 3n n n(3n 1). (This pattern is harder to find intuitively.) So, after she has sampled n trays, she has eaten a total of cookies. n(3n 1) Q3. a. From the first tray, she takes a cookies, from the second, a d cookies and from the third, a d cookies. From the nth tray, she takes a (n 1)d cookies. (Note that these numbers, a, a d, a d, p, a (n 1)d, form an arithmetic sequence with first term a and common difference d.) b. After she has sampled cookies from one tray, she has eaten a cookies. After she has sampled cookies from two trays, she has eaten a (a d) cookies. After she has sampled cookies from three trays, she has eaten a (a d) (a d) cookies. After she has sampled cookies from n trays, she has eaten a total of a (a d) (a d) p (a (n 1)d) cookies. We then simplify as follows: a # n (0 1 # d # d 3 # d p (n 1) # d) n # a d # (1 3 p (n 1)) an d # (n 1)n dn (a d)n. So, after tasting cookies from n trays, she has eaten dn (a d)n cookies. (Note that the numbers a, a d, dn (a d)n 3a 3d, p, form an arithmetic series.)

7 Additional Notes from Connections (technology connection) Using a calculator with a quadratic regression package allows students to quickly find the formulas of the general term of an arithmetic series. For example, letting x be the tray number and y be the total number of cookies eaten, in (1b) we have: after tray number p x total number of p y cookies eaten Using quadratic regression with the TI-83 yields a perfect quadratic fit (R 1), for y ax bx c where a 1, b, c 0. So, in other words, we have y x x x(x ), as above. Similarly, in (b) we have: after tray number p x total number of p y cookies eaten And again, using quadratic regression with the TI-83 yields a perfect quadratic fit for y ax bx c where a 1.5, b 0.5, c 0. So, in this case, we have y 1.5x 0.5x 3x x, as above.