# Standard 22 Identify arithmetic sequences Tell whether the sequence is arithmetic. a. –4, 1, 6, 11, 16,... b. 3, 5, 9, 15, 23,... SOLUTION Find the differences.

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Standard 22 Identify arithmetic sequences Tell whether the sequence is arithmetic. a. –4, 1, 6, 11, 16,... b. 3, 5, 9, 15, 23,... SOLUTION Find the differences of consecutive terms. a 2 – a 1 = 1 – (–4) = 5 a. a 3 – a 2 = 6 – 1 = 5 a 4 – a 3 = 11 – 6 = 5 a 5 – a 4 = 16 – 11 = 5 b. a 2 – a 1 = 5 – 3 = 2 a 3 – a 2 = 9 – 5 = 4 a 4 – a 3 = 15 – 9 = 6 a 5 – a 4 = 23 – 15 = 8

EXAMPLE 1 Identify arithmetic sequences Each difference is 5, so the sequence is arithmetic. ANSWER The differences are not constant, so the sequence is not arithmetic.

GUIDED PRACTICE for Example 1 1. Tell whether the sequence 17, 14, 11, 8, 5,... is arithmetic. Explain why or why not. ANSWER Arithmetic; There is a common differences of –3

EXAMPLE 2 Write a rule for the nth term a. 4, 9, 14, 19,... b. 60, 52, 44, 36,... SOLUTION The sequence is arithmetic with first term a 1 = 4 and common difference d = 9 – 4 = 5. So, a rule for the nth term is: a n = a 1 + (n – 1) d = 4 + (n – 1)5 = –1 + 5n Write general rule. Substitute 4 for a 1 and 5 for d. Simplify. The 15th term is a 15 = –1 + 5(15) = 74. Write a rule for the nth term of the sequence. Then find a 15. a.

EXAMPLE 2 Write a rule for the nth term The sequence is arithmetic with first term a 1 = 60 and common difference d = 52 – 60 = –8. So, a rule for the nth term is: a n = a 1 + (n – 1) d = 60 + (n – 1)(–8) = 68 – 8n Write general rule. Substitute 60 for a 1 and – 8 for d. Simplify. b. The 15th term is a 15 = 68 – 8(15) = –52.

EXAMPLE 3 Write a rule given a term and common difference One term of an arithmetic sequence is a 19 = 48. The common difference is d = 3. a n = a 1 + (n – 1)d a 19 = a 1 + (19 – 1)d 48 = a 1 + 18(3) Write general rule. Substitute 19 for n Solve for a 1. So, a rule for the n th term is: a. Write a rule for the nth term. b. Graph the sequence. –6 = a 1 Substitute 48 for a 19 and 3 for d. SOLUTION a. Use the general rule to find the first term.

EXAMPLE 3 Write a rule given a term and common difference a n = a 1 + (n – 1)d = –6 + (n – 1)3 = –9 + 3n Write general rule. Substitute –6 for a 1 and 3 for d. Simplify. Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence. b.

EXAMPLE 4 Write a rule given two terms Two terms of an arithmetic sequence are a 8 = 21 and a 27 = 97. Find a rule for the nth term. SOLUTION STEP 1 Write a system of equations using a n = a 1 + (n – 1)d and substituting 27 for n (Equation 1 ) and then 8 for n (Equation 2 ).

EXAMPLE 4 Write a rule given two terms STEP 2 Solve the system. 76 = 19d 4 = d 97 = a 1 + 26(4) Subtract. Solve for d. Substitute for d in Equation 1. –7 = a 1 Solve for a 1. STEP 3 Find a rule for a n. a n = a 1 + (n – 1)d Write general rule. = –7 + (n – 1)4 Substitute for a 1 and d. = –11 + 4n Simplify. a 27 = a 1 + (27 – 1)d 97 = a 1 + 26d a 8 = a 1 + (8 – 1)d 21 = a 1 + 7d Equation 1 Equation 2

GUIDED PRACTICE for Examples 2, 3, and 4 Write a rule for the nth term of the arithmetic sequence. Then find a 20. 2. 17, 14, 11, 8,... ANSWER a n = 20 – 3n; –40 3. a 11 = –57, d = –7 ANSWER a n = 20 – 7n; –120 4. a 7 = 26, a 16 = 71 ANSWER a n = –9 + 5n; 91

GUIDED PRACTICE Arithmetic Series The sum of the first n terms of an arithmetic series is given by:

EXAMPLE 5 Standardized Test Practice SOLUTION a 1 = 3 + 5(1) = 8 a 20 = 3 + 5(20) =103 S 20 = 20 ( ) 8 + 103 2 = 1110 Identify first term. Identify last term. Write rule for S 20, substituting 8 for a 1 and 103 for a 20. Simplify. ANSWER The correct answer is C.

GUIDED PRACTICE for Examples 5 and 6 5. Find the sum of the arithmetic series (2 + 7i). 12 i = 1 ANSWER S 12 = 570 ANSWER 610

HOMEWORK Standard 22 Homework PH book page #622 Problems 1-26 all.

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