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.

a2 – a1 = 1 – (–4) = 5a.

a3 – a2 = 6 – 1 = 5

a4 – a3 = 11 – 6 = 5

a5 – a4 = 16 – 11 = 5

b. a2 – a1 = 5 – 3 = 2

a3 – a2 = 9 – 5 = 4

a4 – a3 = 15 – 9 = 6

a5 – a4 = 23 – 15 = 8

EXAMPLE 1 Identify arithmetic sequences

Each difference is 5, so the sequence is arithmetic.

ANSWER 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 a1 = 4 and common difference d = 9 – 4 = 5. So, a rule for the nth term is:an = a1 + (n – 1) d

= 4 + (n – 1)5

= –1 + 5n

Write general rule.

Substitute 4 for a1 and 5 for d.

Simplify.

The 15th term is a15 = –1 + 5(15) = 74.

Write a rule for the nth term of the sequence. Then find a15.

a.

EXAMPLE 2 Write a rule for the nth term

The sequence is arithmetic with first term a1 = 60 and common difference d = 52 – 60 = –8. So, a rule for the nth term is:

an = a1 + (n – 1) d

= 60 + (n – 1)(–8)

= 68 – 8n

Write general rule.

Substitute 60 for a1 and – 8 for d.

Simplify.

b.

The 15th term is a15 = 68 – 8(15) = –52.

EXAMPLE 3 Write a rule given a term and common difference

One term of an arithmetic sequence is a19 = 48. The common difference is d = 3.

an = a1 + (n – 1)d

a19 = a1 + (19 – 1)d

48 = a1 + 18(3)

Write general rule.

Substitute 19 for n

Solve for a1.

So, a rule for the nth term is:

a. Write a rule for the nth term. b. Graph the sequence.

–6 = a1

Substitute 48 for a19 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

an = a1 + (n – 1)d

= –6 + (n – 1)3= –9 + 3n

Write general rule.

Substitute –6 for a1 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 a8 = 21 and a27 = 97. Find a rule for the nth term.

SOLUTION

STEP 1

Write a system of equations using an = a1 + (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 = a1 + 26(4)

Subtract.

Solve for d.

Substitute for d in Equation 1.

–7 = a1 Solve for a1.

STEP 3 Find a rule for an. an = a1 + (n – 1)d Write general rule.

= –7 + (n – 1)4 Substitute for a1 and d.

= –11 + 4n Simplify.

a27 = a1 + (27 – 1)d 97 = a1 + 26da8 = a1 + (8 – 1)d 21 = a1 + 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 a20.2. 17, 14, 11, 8, . . .

ANSWER an = 20 – 3n; –40

3. a11 = –57, d = –7

ANSWER an = 20 – 7n; –120

4. a7 = 26, a16 = 71

ANSWER an = –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

a1 = 3 + 5(1) = 8

a20 = 3 + 5(20) =103

S20 = 20 ( )8 + 103 2

= 1110

Identify first term.

Identify last term.

Write rule for S20, substituting 8 for a1 and 103 for a20.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 = 1ANSWER S12 = 570

 

ANSWER 610

HOMEWORK Standard 22

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