10.2 Analyze Arithmetic Sequences and Series
Warm Up
Lesson Presentation
Lesson Quiz
10.2 Warm-Up
How is each term in the sequence related to the
previous term?
1. 0, 3, 6, 9, 12, …
ANSWER
Each is 3 more than the previous term.
2. 13, 8, 3, –2, –7, …
ANSWER
Each is 5 less than the previous term.
10.2 Warm-Up
Write a rule for the nth term of the sequence. Then find a5.
3. 3, 6, 9, 12, …
ANSWER
4.
an = 3n; a5 = 15
–8, –16, –24, –32, …
ANSWER
an = –8n; a5 = –40
10.2 Example 1
Tell whether the sequence is arithmetic.
b. 3, 5, 9, 15, 23, . . .
a. –4, 1, 6, 11, 16, . . .
SOLUTION
Find the differences of consecutive terms.
a. a2 – a1 = 1 – (–4) = 5
b.
a2 – a1 = 5 – 3 = 2
a3 – a2 = 6 – 1 = 5
a3 – a2 = 9 – 5 = 4
a4 – a3 = 11 – 6 = 5
a4 – a3 = 15 – 9 = 6
a5 – a4 = 16 – 11 = 5
a5 – a4 = 23 – 15 = 8
10.2 Example 1
ANSWER
ANSWER
Each difference is 5, so
the sequence is
arithmetic.
The differences are not
constant, so the
sequence is not
arithmetic.
10.2 Guided Practice
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.
10.2 Example 2
a.
Write a rule for the nth term of the sequence.
Then find a15.
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
Write general rule.
= 4 + (n – 1)5
Substitute 4 for a1 and 5 for d.
Simplify.
= –1 + 5n
The 15th term is a15 = –1 + 5(15) = 74.
10.2 Example 2
b.
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
Write general rule.
= 60 + (n – 1)(–8)
Substitute 60 for a1 and – 8 for d.
= 68 – 8n
Simplify.
The 15th term is a15 = 68 – 8(15) = –52.
10.2 Example 3
One term of an arithmetic sequence is a19 = 48. The
common difference is d = 3.
a. Write a rule for the nth term. b. Graph the sequence.
SOLUTION
a. Use the general rule to find the first term.
an = a1 + (n – 1)d
Write general rule.
a19 = a1 + (19 – 1)d
Substitute 19 for n
48 = a1 + 18(3)
Substitute 48 for a19 and 3 for d.
–6 = a1
Solve for a1.
So, a rule for the nth term is:
10.2 Example 3
an = a1 + (n – 1)d
= –6 + (n – 1)3
= –9 + 3n
Write general rule.
Substitute –6 for a1 and 3 for d.
Simplify.
b. 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.
10.2 Example 4
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).
10.2 Example 4
a27 = a1 + (27 – 1)d
a8 = a1 + (8 – 1)d
97 = a1 + 26d
21 = a1 + 7d
STEP 2 Solve the system. 76 = 19d
4=d
Equation 1
Equation 2
Subtract.
Solve for d.
97 = a1 + 26(4)
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.
10.2 Guided Practice
Write a rule for the nth term of the arithmetic
sequence. Then find a20.
2. 17, 14, 11, 8, . . .
ANSWER
3.
an = 20 – 3n; –40
a11 = –57, d = –7
ANSWER
an = 20 – 7n; –120
4. a7 = 26, a16 = 71
ANSWER
an = –9 + 5n; 91
10.2 Example 5
SOLUTION
a1 = 3 + 5(1) = 8
Identify first term.
a20 = 3 + 5(20) =103 Identify last term.
(
S20 = 20 8 + 103
2
= 1110
)
Write rule for S20, substituting 8 for a1 and
103 for a20.
Simplify.
ANSWER The correct answer is C.
10.2 Example 6
House Of Cards
You are making a house of
cards similar to the one
shown.
a. Write a rule for the number of
cards in the nth row if the top
row is row 1.
b. What is the total number of
cards if the house of cards
has 14 rows?
10.2 Example 6
SOLUTION
a. Starting with the top row, the numbers of cards in
the rows are 3, 6, 9, 12, . . . . These numbers form
an arithmetic sequence with a first term of 3 and a
common difference of 3. So, a rule for the
sequence is:
an = a1 + (n – 1) = d Write general rule.
= 3 + (n – 1)3
= 3n
Substitute 3 for a1 and 3 for d.
Simplify.
10.2 Example 6
SOLUTION
b. Find the sum of an arithmetic series with first
term a1 = 3 and last term a14 = 3(14) = 42.
Total number of cards = S14 = 14
(
a1 + a14
2
) = 14( 3 +242 ) = 315
10.2 Guided Practice
12
Find the sum of the arithmetic series
(2 + 7i).
i=1
S12 = 570
ANSWER
5.
6.
WHAT IF? In Example 6, what is the total number
of cards if the house of cards has 8 rows?
ANSWER
108 cards
10.2 Lesson Quiz
1. Is the sequence 2, 103, 204, 305, 406, . . . arithmetic?
Explain your answer.
ANSWER Yes; the common difference is 101.
2. Write a rule for the nth term of the sequence 5, 2, –1,
–4, . . .. Then find a5.
ANSWER an = 8 – 3n ; a5 = –7
3.
Two terms of an arithmetic sequence are a5 = 14
and a30 = 89. Find a rule for the nth term.
ANSWER an = –1 + 3n
10.2 Lesson Quiz
A movie theater has 24 seats in the first row and each
successive row contains one additional seat. There
are 30 rows in all.
4. Write a rule for the number of seats in the nth row.
ANSWER an = 23 + n
5.
How many seats are in the theater?
ANSWER 1155 seats