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, 3.125 108, 115 17, 15

SLO to identify and find terms in an arithmetic sequence and write the sequence in an explicit formula

Vocabulary sequence term arithmetic sequence common difference

A sequence is an ordered list of numbers or objects A sequence is an ordered list of numbers or objects. Each number of object in a sequence is called a term. 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.

You cannot tell if a sequence is arithmetic by looking at a finite number of terms because the next term might not fit the pattern. Caution!

Additional Example 1A: Identifying Arithmetic Sequences Determine if the sequence could be arithmetic. If so, give the common difference. 5, 8, 11, 14, 17, . . . The terms increase by 3. 5 8 11 14 17, . . . +3 +3 +3 +3 The sequence could be arithmetic with a common difference of 3.

Additional Example 1B: Identifying Arithmetic Sequences Determine if the sequence could be arithmetic. If so, give the common difference. 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.

Check It Out: Example 1A Determine if the sequence could be arithmetic. If so, give the common difference. 1, 2, 3, 4, 5, . . . The terms increase by 1. 1 2 3 4 5, . . . +1 +1 +1 +1 The sequence could be arithmetic with a common difference of 1.

Check It Out: Example 1B Determine if the sequence could be arithmetic. If so, give the common difference. 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 2: Finding Missing Terms in an Arithmetic Sequence Find the next three terms in the arithmetic sequence –8, –3, 2, 7, ... Each term is 5 more than the previous term. 7 + 5 = 12 Use the common difference to find the next three terms. 12 + 5 = 17 17 + 5 = 22 The next three terms are 12, 17, and 22.

Check It Out: Example 2 Find the next three terms in the arithmetic sequence –9, –6, –3, 0, ... Each term is 3 more than the previous term. 0 + 3 = 3 Use the common difference to find the next three terms. 3 + 3 = 6 6 + 3 = 9 The next three terms are 3, 6, and 9.

Additional Example 3A: Identifying Functions in Arithmetic Sequences Find a function that describes each arithmetic sequence. Use y to identify each term in the sequence and n to identify each term’s position. 6, 12, 18, 24, … n n • 6 y 1 2 3 4 1 • 6 6 Multiply n by 6. 2 • 6 12 3 • 6 18 y = 6n 4 • 6 24 n • 6 6n

Additional Example 3B: Identifying Functions in Arithmetic Sequences Find a function that describes each arithmetic sequence. Use y to identify each term in the sequence and n to identify each term’s position. –4, –8, –12, –16, … n n • (– 4) y 1 2 3 4 1 • (–4) –4 Multiply n by –4. 2 • (–4) –8 3 • (–4) –12 y = –4n 4 • (–4) –16 n • (–4) –4n

Additional Example 4A: Finding a Given Term of an Arithmetic Sequence Find the given term in the arithmetic sequence. 10th term: 1, 3, 5, 7, . . . an = a1 + (n – 1)d a10 = 1 + (10 – 1)2 a10 = 19

Additional Example 4B: Finding a Given Term of an Arithmetic Sequence Find the given term in the arithmetic sequence. 18th term: 100, 93, 86, 79, . . . an = a1 + (n – 1)d a18 = 100 + (18 – 1)(–7) a18 = –19

Check It Out: Example 4A Find the given term in the arithmetic sequence. 15th term: 1, 3, 5, 7, . . . an = a1 + (n – 1)d a15 = 1 + (15 – 1)2 a15 = 29

an = a1 + (n – 1)d a50 = 100 + (50 – 1)(–7) a50 = –243 Check It Out: Example 4B Find the given term in the arithmetic sequence. 50th term: 100, 93, 86, 79, . . . an = a1 + (n – 1)d a50 = 100 + (50 – 1)(–7) a50 = –243

Additional Example 5: 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 a1 = 20.5 = money after first sale d = 0.5 d = .50 = common difference an = 63.5 an = 63.5 = money at the end of the sale

Additional Example 5 Continued Let n represent the item number of cookies sold that will earn the class a total of $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.

Check It Out: Example 5 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 a1 = 10.25 = money after first sale d = 0.25 d = .25 = common difference an = 40 an = 40 = money at the end of the sale

Check It Out: Example 5 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.

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 23

Lesson Quiz: Part I 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 next three terms each arithmetic sequence. 3. 18, 13, 8, 3, … 4. 3.6, 5, 6.4, 7.8, … yes; 7 no –2, –7, –12 9.2, 10.6, 12

Lesson Quiz: Part II Find the given term in each arithmetic sequence. 5. 15th term: a1 = 7, d = 5 6. 24th term: 1, , , , 2 77 5 4 3 2 7 4 , or 6.75 27 4

Lesson Quiz for Student Response Systems 1. Determine if each sequence could be arithmetic. If so, give the common difference. 15, 18, 21, 24, 27, 30, … A. yes; +3 B. yes; –3 C. no; +3 D. no; –3 26

Lesson Quiz for Student Response Systems 2. Find the next three terms in the arithemtic sequence. 15, 11, 7, 3, –1, –5, … A. –9, –12, –16 B. –9, –13, –17 C. 9, 13, 17 D. 9, 12, 16 27

Lesson Quiz for Student Response Systems 3. Find the function that describes the arithmetic sequence. 15, 11, 7, 3, –1, –5, … A. y = n – 3 B. y = n – 4 C. y = 3n D. y = 4n 28

Lesson Quiz for Student Response Systems 4. Find the 7th term in the arithmetic sequence. 1, 3, 5, 7, … A. 9 B. 13 C. 15 D. 49 29