Download presentation

Presentation is loading. Please wait.

Published byAllen Fields Modified over 8 years ago

1
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

2
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.

3
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

4
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.

5
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.

6
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.

7
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.

8
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 ).

9
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

10
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

11
GUIDED PRACTICE Arithmetic Series The sum of the first n terms of an arithmetic series is given by:

12
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.

13
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

14
HOMEWORK Standard 22 Homework PH book page #622 Problems 1-26 all.

Similar presentations

© 2024 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google