Warm Up Find the next two numbers in the pattern, using the simplest rule you can find. 1. 1, 5, 9, 13, . . . 2. 100, 50, 25, 12.5, . . . 3. 80, 87, 94, 101, . . . 4. 3, 9, 7, 13, 11, . . . 17, 21 6.25, 3.125 45 Bus 2 Ahmedi Abdulkahar 25014283 Arango Laura 23878556 Blackman Tia 22775662 Bohn Sebastian 25937806 Burkett Jayson 22894430 Cassisa Dominic 22287155 Castellano Joe 22811178 Coban John 22791826 Correggio Joe 26139840 Da Motta Livia 26308130 10 Dasgupta Riyana 25764523 Dionne Danielle 24027369 Duque Michelle 23721376 Gerzina Lindsey 23234438 Gordon Taylor 26140319 Gordon Macie 26164921 Kibler Steven 22813885 Lara Ariana 23299084 Long Shuangxou 25448754 Martins Yves 26192260 20 Mendez Andres 25828765 Midyette Angela 24159287 Miller Zachary 22763494 Narpiel Brandon 25404039 Nathan Jordan 25420712 Nelson Shannon 24977456 Ollestad Stefan 25826777 Orlowski Cristina 22801658 Pollack Leah 24510430 Pomerantz Jessica 23827264 30 Rhodes Adam 24959835 Roth Bailey 22845994 Rucco Dominic 22818249 Salamanca Oscar 25177387 Secatello Amanda 24142002 Serafini Nicole 22950372 Silver Oliver 24059313 37 Sincoff Danielle 22860886 Towbin Joey 22814024 Vela Melissa 22937593 Walker Kirsten 22807200 Webb Katarina 25203407 Wiernicki Richard 22813885 Woods Ashley 22814263 Zhou Kelly 25439472 45 108, 115 17, 15

Arithmetic Sequences 12.1 Pre-Algebra

Learn to find terms in an arithmetic sequence.

Vocabulary sequence term arithmetic sequence common difference

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.

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

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

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

Example: 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, . . . Find the difference of each term and the term before it. 5.7 5.8 5.9 6 6.1, . . . 0.1 0.1 0.1 0.1 The sequence could be arithmetic with a common difference of 0.1.

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

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

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

Try This Determine if the sequence could be arithmetic. If so, give the common difference. C. 11, 22, 33, 44, 55, . . . Find the difference of each term and the term before it. 11 22 33 44 55, . . . 11 11 11 11 The sequence could be arithmetic with a common difference of 11.

Try This Determine if the sequence could be arithmetic. If so, give the common difference. D. 1, 1, 1, 1, 1, 1, . . . Find the difference of each term and the term before it. 1 1 1 1 1, . . . The sequence could be arithmetic with a common difference of 0.

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

The nth term an of an arithmetic sequence with common difference d is Writing Math Subscripts are used to show the positions of terms in the sequence. The first term is a1, the second is a2, 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 = a1 + (n – 1)d. The term we are looking for. The FIRST term in the series. The NUMBER of the term we are looking for. The COMMON DIFFERENCE.

Example an = a1 + (n – 1)d a10 = 1 + (10 – 1)2 a10 = 19 Find the given term in the arithmetic sequence. A. 10th term: 1, 3, 5, 7, . . . an = a1 + (n – 1)d a10 = 1 + (10 – 1)2 a10 = 19

Example an = a1 + (n – 1)d a18 = 100 + (18 – 1)(–7) a18 = -19 Find the given term in the arithmetic sequence. B. 18th term: 100, 93, 86, 79, . . . an = a1 + (n – 1)d a18 = 100 + (18 – 1)(–7) a18 = -19

Example an = a1 + (n – 1)d a21 = 25 + (21 – 1)(0.5) a21 = 35 Find the given term in the arithmetic sequence. C. 21st term: 25, 25.5, 26, 26.5, . . . an = a1 + (n – 1)d a21 = 25 + (21 – 1)(0.5) a21 = 35

Example an = a1 + (n – 1)d a14 = 13 + (14 – 1)5 a14 = 78 Find the given term in the arithmetic sequence. D. 14th term: a1 = 13, d = 5 an = a1 + (n – 1)d a14 = 13 + (14 – 1)5 a14 = 78

Try This an = a1 + (n – 1)d a15 = 1 + (15 – 1)2 a15 = 29 Find the given term in the arithmetic sequence. A. 15th term: 1, 3, 5, 7, . . . an = a1 + (n – 1)d a15 = 1 + (15 – 1)2 a15 = 29

Try This an = a1 + (n – 1)d a50 = 100 + (50 – 1)(-7) a50 = –243 Find the given term in the arithmetic sequence. B. 50th term: 100, 93, 86, 79, . . . an = a1 + (n – 1)d a50 = 100 + (50 – 1)(-7) a50 = –243

Try This an = a1 + (n – 1)d a41 = 25 + (41 – 1)(0.5) a41 = 45 Find the given term in the arithmetic sequence. C. 41st term: 25, 25.5, 26, 26.5, . . . an = a1 + (n – 1)d a41 = 25 + (41 – 1)(0.5) a41 = 45

Try This an = a1 + (n – 1)d a2 = 13 + (2 – 1)5 a2 = 18 Find the given term in the arithmetic sequence. D. 2nd term: a1 = 13, d = 5 an = a1 + (n – 1)d a2 = 13 + (2 – 1)5 a2 = 18

You can use the formula for the nth term of an arithmetic sequence to solve for other variables.

Example 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, 21.5, 22, . . . a1 = 20.5 Let a1 = 20.5 = money after first sale. d = 0.5 an = 63.5

Example Let n represent the item number in which the cash box will contain $63.50. Use the formula for arithmetic sequences. an = a1 + (n – 1) d 63.5 = 20.5 + (n – 1)(0.5) Solve for n. 63.5 = 20.5 + 0.5n – 0.5 Distributive Property. 63.5 = 20 + 0.5n Combine like terms. 43.5 = 0.5n Subtract 20 from both sides. 87 = n Divide both sides by 0.5. During the bake sale, 87 items are sold in order for the cash box to contain $63.50.

Try This 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, 10.5, 10.75, 11, … a1 = 10.25 Let a1 = 10.25 = money after first sale. d = 0.25 an = 40

Try This Let n represent the number of pencils in which he will have $40 in his money bag. Use the formula for arithmetic sequences. an = a1 + (n – 1)d 40 = 10.25 + (n – 1)(0.25) Solve for n. 40 = 10.25 + 0.25n – 0.25 Distributive Property. Combine like terms. 40 = 10 + 0.25n 30 = 0.25n 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.

Lesson Quiz Determine if each sequence could be arithmetic. If so, give the common difference. 1. 42, 49, 56, 63, 70, . . . 2. 1, 2, 4, 8, 16, 32, . . . Find the given term in each arithmetic sequence. 3. 15th term: a1 = 7, d = 5 4. 24th term: 1, , , , 2 5. 52nd term: a1 = 14.2; d = –1.2 yes; 7 no 77 5 4 3 2 , or 6.75 27 4 7 4 –47