standard 22 identify arithmetic sequences tell whether the sequence is arithmetic. a. –4, 1, 6,...
TRANSCRIPT
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.