Arithmetic Sequences 4.6, Summaries of Algebra

Download Arithmetic Sequences 4.6 and more Summaries Algebra in PDF only on Docsity! Section 4.6 Arithmetic Sequences 209 Arithmetic Sequences4.6 Describing a Pattern Work with a partner. Use the fi gures to complete the table. Plot the points given by your completed table. Describe the pattern of the y-values. a. n = 1 n = 2 n = 3 n = 4 n = 5 Number of stars, n 1 2 3 4 5 Number of sides, y b. n = 1 n = 2 n = 3 n = 4 n = 5 n 1 2 3 4 5 Number of circles, y c. n = 1 n = 2 n = 3 n = 4 n = 5 Number of rows, n 1 2 3 4 5 Number of dots, y Communicate Your Answer 2. How can you use an arithmetic sequence to describe a pattern? Give an example from real life. 3. In chemistry, water is called H2O because each molecule of water has two hydrogen atoms and one oxygen atom. Describe the pattern shown below. Use the pattern to determine the number of atoms in 23 molecules. n = 1 n = 2 n = 3 n = 4 n = 5 Essential Question How can you use an arithmetic sequence to describe a pattern? An arithmetic sequence is an ordered list of numbers in which the difference between each pair of consecutive terms, or numbers in the list, is the same. LOOKING FOR A PATTERN To be profi cient in math, you need to look closely to discern patterns and structure. n y 1 2 3 4 50 0 10 20 30 40 50 60 n y 1 2 3 4 50 0 1 2 3 4 5 6 n y 1 2 3 4 50 0 2 4 6 8 10 12 hsnb_alg1_pe_0406.indd 209 2/4/15 4:03 PM 210 Chapter 4 Writing Linear Functions 4.6 Lesson What You Will Learn Write the terms of arithmetic sequences. Graph arithmetic sequences. Write arithmetic sequences as functions. Writing the Terms of Arithmetic Sequences A sequence is an ordered list of numbers. Each number in a sequence is called a term. Each term an has a specifi c position n in the sequence. 5, 10, 15, 20, 25, . . . , an, . . . sequence, p. 210 term, p. 210 arithmetic sequence, p. 210 common difference, p. 210 Previous point-slope form function notation Core Vocabulary Extending an Arithmetic Sequence Write the next three terms of the arithmetic sequence. −7, −14, −21, −28, . . . SOLUTION Use a table to organize the terms and fi nd the pattern. Add −7 to a term to fi nd the next term. The next three terms are −35, −42, and −49. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write the next three terms of the arithmetic sequence. 1. −12, 0, 12, 24, . . . 2. 0.2, 0.6, 1, 1.4, . . . 3. 4, 3 3 — 4 , 3 1 — 2 , 3 1 — 4 , . . . READING An ellipsis (. . .) is a series of dots that indicates an intentional omission of information. In mathematics, the . . . notation means “and so forth.” The ellipsis indicates that there are more terms in the sequence that are not shown. Core Concept Arithmetic Sequence In an arithmetic sequence, the difference between each pair of consecutive terms is the same. This difference is called the common difference. Each term is found by adding the common difference to the previous term. 5, 10, 15, 20, . . . Terms of an arithmetic sequence +5 +5 +5 common difference 1st position 3rd position nth position Each term is 7 less than the previous term. So, the common difference is −7. Position 1 2 3 4 Term −7 −14 −21 −28 +(−7) +(−7) +(−7) Position 1 2 3 4 5 6 7 Term −7 −14 −21 −28 −35 −42 −49 +(−7) +(−7) +(−7) hsnb_alg1_pe_0406.indd 210 2/4/15 4:03 PM Section 4.6 Arithmetic Sequences 213 You can rewrite the equation for an arithmetic sequence with fi rst term a1 and common difference d in function notation by replacing an with f(n). f(n) = a1 + (n − 1)d The domain of the function is the set of positive integers. Writing Real-Life Functions Online bidding for a purse increases by $5 for each bid after the $60 initial bid. a. Write a function that represents the arithmetic sequence. b. Graph the function. c. The winning bid is $105. How many bids were there? SOLUTION a. The fi rst term is 60, and the common difference is 5. f(n) = a1 + (n − 1)d Function for an arithmetic sequence f(n) = 60 + (n − 1)5 Substitute 60 for a1 and 5 for d. f(n) = 5n + 55 Simplify. The function f(n) = 5n + 55 represents the arithmetic sequence. b. Make a table. Then plot the ordered pairs (n, an). Bid number, n Bid amount, an 1 60 2 65 3 70 4 75 0 0B id a m o u n t (d o lla rs ) n an Bid number Bidding on a Purse 1 2 3 4 5 6 55 60 65 70 75 80 (1, 60) (2, 65) (3, 70) (4, 75) c. Use the function to fi nd the value of n for which f(n) = 105. f(n) = 5n + 55 Write the function. 105 = 5n + 55 Substitute 105 for f(n). 10 = n Solve for n. There were 10 bids. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 11. A carnival charges $2 for each game after you pay a $5 entry fee. a. Write a function that represents the arithmetic sequence. b. Graph the function. c. How many games can you play when you take $29 to the carnival? Games Total cost 1 $7 2 $9 3 $11 4 $13 REMEMBER The domain is the set of positive integers. Bid number 1 2 3 4 Bid amount $60 $65 $70 $75 hsnb_alg1_pe_0406.indd 213 2/4/15 4:03 PM 214 Chapter 4 Writing Linear Functions Exercises4.6 Dynamic Solutions available at BigIdeasMath.com In Exercises 3 and 4, write the next three terms of the arithmetic sequence. 3. First term: 2 Common difference: 13 4. First term: 18 Common difference: −6 In Exercises 5−10, fi nd the common difference of the arithmetic sequence. 5. 13, 18, 23, 28, . . . 6. 175, 150, 125, 100, . . . 7. −16, −12, −8, −4, . . . 8. 4, 3 2 — 3 , 3 1 — 3 , 3, . . . 9. 6.5, 5, 3.5, 2, . . . 10. −16, −7, 2, 11, . . . In Exercises 11−16, write the next three terms of the arithmetic sequence. (See Example 1.) 11. 19, 22, 25, 28, . . . 12. 1, 12, 23, 34, . . . 13. 16, 21, 26, 31, . . . 14. 60, 30, 0, −30, . . . 15. 1.3, 1, 0.7, 0.4, . . . 16. 5 — 6 , 2 — 3 , 1 — 2 , 1 — 3 , . . . In Exercises 17−22, graph the arithmetic sequence. (See Example 2.) 17. 4, 12, 20, 28, . . . 18. −15, 0, 15, 30, . . . 19. −1, −3, −5, −7, . . . 20. 2, 19, 36, 53, . . . 21. 0, 4 1 — 2 , 9, 13 1 — 2 , . . . 22. 6, 5.25, 4.5, 3.75, . . . In Exercises 23−26, determine whether the graph represents an arithmetic sequence. Explain. (See Example 3.) 23. n an 1 2 3 4 5 6 70 0 1 2 3 4 5 6 7 (1, 1) (3, 1) (2, 4) (4, 4) 24. n an 1 2 3 4 5 6 70 0 4 8 12 16 20 24 28 (1, 5) (3, 19) (2, 12) (4, 26) 25. n an 1 2 3 4 5 6 70 0 10 20 30 40 50 60 70 (1, 70) (3, 40) (2, 55) (4, 25) 26. n an 1 2 3 4 5 6 70 0 3 6 9 12 15 18 21 (2, 10) (1, 2) (3, 16) (4, 20) In Exercises 27−30, determine whether the sequence is arithmetic. If so, fi nd the common difference. 27. 13, 26, 39, 52, . . . 28. 5, 9, 14, 20, . . . 29. 48, 24, 12, 6, . . . 30. 87, 81, 75, 69, . . . 31. FINDING A PATTERN Write a sequence that represents the number of smiley faces in each group. Is the sequence arithmetic? Explain. Monitoring Progress and Modeling with Mathematics 1. WRITING Describe the graph of an arithmetic sequence. 2. DIFFERENT WORDS, SAME QUESTION Consider the arithmetic sequence represented by the graph. Which is different? Find “both” answers. Find the slope of the linear function. Find the difference between consecutive terms of the arithmetic sequence. Find the difference between the terms a2 and a4. Find the common difference of the arithmetic sequence. Vocabulary and Core Concept Check n an 1 2 3 4 5 6 70 0 3 6 9 12 15 18 21 (2, 10) (3, 13) (4, 16) (5, 19) hsnb_alg1_pe_0406.indd 214 2/4/15 4:03 PM Section 4.6 Arithmetic Sequences 215 32. FINDING A PATTERN Write a sequence that represents the sum of the numbers in each roll. Is the sequence arithmetic? Explain. Roll 1 Roll 2 Roll 3 Roll 4 In Exercises 33−38, write an equation for the nth term of the arithmetic sequence. Then fi nd a10. (See Example 4.) 33. −5, −4, −3, −2, . . . 34. −6, −9, −12, −15, . . . 35. 1 — 2 , 1, 1 1 — 2 , 2, . . . 36. 100, 110, 120, 130, . . . 37. 10, 0, −10, −20, . . . 38. 3 — 7 , 4 — 7 , 5 — 7 , 6 — 7 , . . . 39. ERROR ANALYSIS Describe and correct the error in fi nding the common difference of the arithmetic sequence. 2, 1, 0, −1, . . . −1 −1 −1 The common diff erence is 1. ✗ 40. ERROR ANALYSIS Describe and correct the error in writing an equation for the nth term of the arithmetic sequence. 14, 22, 30, 38, . . . an = a1 + nd an = 14 + 8n ✗ 41. NUMBER SENSE The fi rst term of an arithmetic sequence is 3. The common difference of the sequence is 1.5 times the fi rst term. Write the next three terms of the sequence. Then graph the sequence. 42. NUMBER SENSE The fi rst row of a dominoes display has 10 dominoes. Each row after the fi rst has two more dominoes than the row before it. Write the fi rst fi ve terms of the sequence that represents the number of dominoes in each row. Then graph the sequence. REPEATED REASONING In Exercises 43 and 44, (a) draw the next three fi gures in the sequence and (b) describe the 20th fi gure in the sequence. 43. 44. 45. MODELING WITH MATHEMATICS The total number of babies born in a country each minute after midnight January 1st can be estimated by the sequence shown in the table. (See Example 5.) Minutes after midnight January 1st 1 2 3 4 Total babies born 5 10 15 20 a. Write a function that represents the arithmetic sequence. b. Graph the function. c. Estimate how many minutes after midnight January 1st it takes for 100 babies to be born. 46. MODELING WITH MATHEMATICS The amount of money a movie earns each week after its release can be approximated by the sequence shown in the graph. 0 0 Ea rn in g s (m ill io n s o f d o lla rs ) n an Week Movie Earnings 1 2 3 4 5 10 20 30 40 50 60 (1, 56) (2, 48) (3, 40) (4, 32) a. Write a function that represents the arithmetic sequence. b. In what week does the movie earn $16 million? c. How much money does the movie earn overall? hsnb_alg1_pe_0406.indd 215 2/4/15 4:03 PM