12 1 Arithmetic Sequences Objective Find terms in

12 -1 Arithmetic Sequences Objective: Find terms in an arithmetic sequence. Pre-Algebra

12 -1 Arithmetic Sequences Vocabulary sequence term arithmetic sequence common difference Pre-Algebra

12 -1 Arithmetic Sequences A sequence is a list of numbers or objects, called terms, in a certain order. In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called the common difference. The common difference is added to each term to get the next term. Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 1 A: Identifying Arithmetic Sequences Determine if the sequence could be arithmetic. If so, give the common difference. A. 5, 8, 11, 14, 17, . . . 5 8 3 11 3 14 3 17, . . . Find the difference of each term and the term before it. 3 The sequence could be arithmetic with a common difference of 3. Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 1 A Determine if the sequence could be arithmetic. If so, give the common difference. A. 1, 2, 3, 4, 5, . . . 1 2 1 3 1 4 1 5, . . . Find the difference of each term and the term before it. 1 The sequence could be arithmetic with a common difference of 1. Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 1 B: Identifying Arithmetic Sequences Determine if the sequence could be arithmetic. If so, give the common difference. B. 1, 3, 6, 10, 15, . . . 1 3 2 6 3 10 4 15, . . . Find the difference of each term and the term before it. 5 The sequence is not arithmetic. Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 1 B Determine if the sequence could be arithmetic. If so, give the common difference. B. 1, 3, 7, 8, 12, … 1 3 2 7 4 8 1 12, . . . Find the difference of each term and the term before it. 4 The sequence is not arithmetic. Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 1 C: Identifying Arithmetic Sequences Determine if the sequence could be arithmetic. If so, give the common difference. C. 65, 60, 55, 50, 45, . . . 65 60 – 5 55 – 5 50 – 5 45, . . . Find the difference of each term and the term before it. – 5 The sequence could be arithmetic with a common difference of – 5. Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 1 C Determine if the sequence could be arithmetic. If so, give the common difference. C. 11, 22, 33, 44, 55, . . . 11 22 11 33 11 11 55, . . . Find the difference of each term and the term before it. 11 44 The sequence could be arithmetic with a common difference of 11. Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 1 D: Identifying Arithmetic Sequences Determine if the sequence could be arithmetic. If so, give the common difference. D. 5. 7, 5. 8, 5. 9, 6, 6. 1, . . . 5. 7 5. 8 0. 1 5. 9 0. 1 6 6. 1, . . . Find the difference of each term and the term before it. 0. 1 The sequence could be arithmetic with a common difference of 0. 1. Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 1 D Determine if the sequence could be arithmetic. If so, give the common difference. D. 1, 1, 1, . . . 1 1 0 1 0 1, . . . Find the difference of each term and the term before it. 0 The sequence could be arithmetic with a common difference of 0. Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 1 E: Identifying Arithmetic Sequences Determine if the sequence could be arithmetic. If so, give the common difference. E. 1, 0, -1, 0, 1, . . . 1 0 – 1 – 1 0 1 1, . . . 1 Find the difference of each term and the term before it. The sequence is not arithmetic. Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 1 E Determine if the sequence could be arithmetic. If so, give the common difference. E. 2, 4, 6, 8, 9, . . . 2 4 2 6 2 8 2 9, . . . Find the difference of each term and the term before it. 1 The sequence is not arithmetic. Pre-Algebra

12 -1 Arithmetic Sequences Writing Math Subscripts are used to show the positions of terms in the sequence. The first term is a 1, the second is a 2, and so on. FINDING THE nth TERM OF AN ARITHMETIC SEQUENCE The nth term an of an arithmetic sequence with common difference d is an = a 1 + (n – 1)d. Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 2 A: Finding a Given Term of an Arithmetic Sequence Find the given term in the arithmetic sequence. A. 10 th term: 1, 3, 5, 7, . . . an = a 1 + (n – 1)d a 10 = 1 + (10 – 1)2 a 10 = 19 Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 2 A Find the given term in the arithmetic sequence. A. 15 th term: 1, 3, 5, 7, . . . an = a 1 + (n – 1)d a 15 = 1 + (15 – 1)2 a 15 = 29 Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 2 B: Finding a Given Term of an Arithmetic Sequence Find the given term in the arithmetic sequence. B. 18 th term: 100, 93, 86, 79, . . . an = a 1 + (n – 1)d a 18 = 100 + (18 – 1)(– 7) a 18 = -19 Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 2 B Find the given term in the arithmetic sequence. B. 50 th term: 100, 93, 86, 79, . . . an = a 1 + (n – 1)d a 50 = 100 + (50 – 1)(-7) a 50 = – 243 Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 2 C: Finding a Given Term of an Arithmetic Sequence Find the given term in the arithmetic sequence. C. 21 st term: 25, 25. 5, 26. 5, . . . an = a 1 + (n – 1)d a 21 = 25 + (21 – 1)(0. 5) a 21 = 35 Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 2 C Find the given term in the arithmetic sequence. C. 41 st term: 25, 25. 5, 26. 5, . . . an = a 1 + (n – 1)d a 41 = 25 + (41 – 1)(0. 5) a 41 = 45 Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 2 D: Finding a Given Term of an Arithmetic Sequence Find the given term in the arithmetic sequence. D. 14 th term: a 1 = 13, d = 5 an = a 1 + (n – 1)d a 14 = 13 + (14 – 1)5 a 14 = 78 Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 2 D Find the given term in the arithmetic sequence. D. 2 nd term: a 1 = 13, d = 5 an = a 1 + (n – 1)d a 2 = 13 + (2 – 1)5 a 2 = 18 Pre-Algebra

12 -1 Arithmetic Sequences You can use the formula for the nth term of an arithmetic sequence to solve for other variables. Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 3: Application The senior class held a bake sale. At the beginning of the sale, there was $20 in the cash box. Each item in the sale cost 50 cents. At the end of the sale, there was $63. 50 in the cash box. How many items were sold during the bake sale? Identify the arithmetic sequence: 20. 5, 21. 5, 22, . . . a 1 = 20. 5 Let a 1 = 20. 5 = money after first sale. d = 0. 5 an = 63. 5 Pre-Algebra

12 -1 Arithmetic Sequences Additional Example 3 Continued Let n represent the item number in which the cash box will contain $63. 50. Use the formula for arithmetic sequences. an = a 1 + (n – 1) d 63. 5 = 20. 5 + (n – 1)(0. 5) Solve for n. 63. 5 = 20. 5 + 0. 5 n – 0. 5 Distributive Property. 63. 5 = 20 + 0. 5 n Combine like terms. 43. 5 = 0. 5 n Subtract 20 from both sides. Divide both sides by 0. 5. 87 = n During the bake sale, 87 items are sold in order for the cash box to contain $63. 50. Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 3 Johnnie is selling pencils for student council. At the beginning of the day, there was $10 in his money bag. Each pencil costs 25 cents. At the end of the day, he had $40 in his money bag. How many pencils were sold during the day? Identify the arithmetic sequence: 10. 25, 10. 75, 11, … a 1 = 10. 25 Let a 1 = 10. 25 = money after first sale. d = 0. 25 an = 40 Pre-Algebra

12 -1 Arithmetic Sequences Try This: Example 3 Continued Let n represent the number of pencils in which he will have $40 in his money bag. Use the formula for arithmetic sequences. an = a 1 + (n – 1)d 40 = 10. 25 + (n – 1)(0. 25) Solve for n. 40 = 10. 25 + 0. 25 n – 0. 25 Distributive Property. 40 = 10 + 0. 25 n Combine like terms. 30 = 0. 25 n Subtract 10 from both sides. 120 = n Divide both sides by 0. 25. 120 pencils are sold in order for his money bag to contain $40. Pre-Algebra