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

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

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

Pre-Algebra 12-1 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,... Example 1D: Identifying Arithmetic Sequences The sequence could be arithmetic with a common difference of 0.1. Find the difference of each term and the term before it ,

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

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 n th TERM OF AN ARITHMETIC SEQUENCE The n th term a n of an arithmetic sequence with common difference d is a n = a 1 + ( n – 1) d.

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

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

Pre-Algebra 12-1 Arithmetic Sequences Find the given term in the arithmetic sequence. C. 21 st term: 25, 25.5, 26, 26.5,... Example 2C: Finding a Given Term of an Arithmetic Sequence a n = a 1 + ( n – 1) d a 21 = 25 + (21 – 1)(0.5) a 21 = 35

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

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

Pre-Algebra 12-1 Arithmetic Sequences The 8th Grade 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? Example 3: Application Identify the arithmetic sequence: 20.5, 21, 21.5, 22,... a 1 = 20.5 Let a 1 = 20.5 = money after first sale. d = 0.5 a n = 63.5

Pre-Algebra 12-1 Arithmetic Sequences Example 3 Continued Let n represent the item number in which the cash box will contain $ Use the formula for arithmetic sequences. a n = a 1 + ( n – 1) d Solve for n = ( n – 1)(0.5) 63.5 = n – 0.5 Distributive Property = n Combine like terms. 87 = n Subtract 20 from both sides. During the bake sale, 87 items are sold in order for the cash box to contain $ = 0.5 n Divide both sides by 0.5.

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

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

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

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

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

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

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

Pre-Algebra 12-1 Arithmetic Sequences Find the given term in the arithmetic sequence. C. 41 st term: 25, 25.5, 26, 26.5,... Try This: Example 2C a n = a 1 + ( n – 1) d a 41 = 25 + (41 – 1)(0.5) a 41 = 45

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

Pre-Algebra 12-1 Arithmetic Sequences John 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? Try This: Example 3 Identify the arithmetic sequence: 10.25, 10.5, 10.75, 11, … a 1 = Let a 1 = = money after first sale. d = 0.25 a n = 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. a n = a 1 + ( n – 1) d Solve for n. 40 = ( n – 1)(0.25) 40 = n – 0.25 Distributive Property. 40 = n Combine like terms. 120 = n Subtract 10 from both sides. 120 pencils are sold in order for his money bag to contain $ = 0.25 n Divide both sides by 0.25.