Objective: 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.

Additional Example 1A: 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.

Try This: Example 1A 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.

Additional Example 1B: 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.

Try This: Example 1B 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.

Additional Example 1C: 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.

Try This: Example 1C 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.

Additional Example 1D: 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.

Try This: Example 1D 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.

Additional Example 1E: 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: Example 1E 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.

FINDING THE nth TERM OF AN ARITHMETIC SEQUENCE 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.

Additional Example 2A: Finding a Given Term of an Arithmetic Sequence 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

an = a1 + (n – 1)d a15 = 1 + (15 – 1)2 a15 = 29 Try This: Example 2A 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

Additional Example 2B: Finding a Given Term of an Arithmetic Sequence 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

an = a1 + (n – 1)d a50 = 100 + (50 – 1)(-7) a50 = –243 Try This: Example 2B 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

Additional Example 2C: Finding a Given Term of an Arithmetic Sequence 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

an = a1 + (n – 1)d a41 = 25 + (41 – 1)(0.5) a41 = 45 Try This: Example 2C 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

Additional Example 2D: Finding a Given Term of an Arithmetic Sequence 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

an = a1 + (n – 1)d a2 = 13 + (2 – 1)5 a2 = 18 Try This: Example 2D 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.

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

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

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 = 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.