Arithmetic Sequences and Series

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12-3
Arithmetic Sequences and Series
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Find the 5th term of each sequence. 1.
an n 6 2. an 4 n 3. an 3n 4 Write a
possible explicit rule for the nth term of each
sequence. 4. 4, 5, 6, 7, 8, 5. 3, 1, 1, 3,
5, 6.
11
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an n 3
an 2n 5
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Objectives
Find the indicated terms of an arithmetic
sequence. Find the sums of arithmetic series.
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Vocabulary
arithmetic sequence arithmetic series
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The cost of mailing a letter in 2005 gives the
sequence 0.37, 0.60, 0.83, 1.06, . This
sequence is called an arithmetic sequence because
its successive terms differ by the same number d
(d ? 0), called the common difference. For the
mail costs, d is 0.23, as shown.
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Recall that linear functions have a constant
first difference. Notice also that when you graph
the ordered pairs (n, an) of an arithmetic
sequence, the points lie on a straight line.
Thus, you can think of an arithmetic sequence as
a linear function with sequential natural numbers
as the domain.
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Check It Out! Example 1a
Determine whether the sequence could be
arithmetic. If so, find the common difference and
the next term.
1.9, 1.2, 0.5, 0.2, 0.9, ...
1.9, 1.2, 0.5, 0.2, 0.9
The sequence could be arithmetic with a common
difference of 0.7.
The next term would be 0.9 0.7 1.6.
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Check It Out! Example 1b
Determine whether the sequence could be
arithmetic. If so, find the common difference and
the next term.
Differences
The sequence is not arithmetic because the first
differences are not common.
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Each term in an arithmetic sequence is the sum of
the previous term and the common difference. This
gives the recursive rule an an 1 d. You
also can develop an explicit rule for an
arithmetic sequence.
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Notice the pattern in the table. Each term is the
sum of the first term and a multiple of the
common difference. This pattern can be
generalized into a rule for all arithmetic
sequences.
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Check It Out! Example 2a
Find the 11th term of the arithmetic sequence.
3, 5, 7, 9,
Step 1 Find the common difference d 5 (3)
2.
Step 2 Evaluate by using the formula.
an a1 (n 1)d
General rule.
Substitute 3 for a1, 11 for n, and 2 for d.
a11 3 (11 1)(2)
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The 11th term is 23.
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Check It Out! Example 2a Continued
Check Continue the sequence.
n 1 2 3 4 5 6 7 8 9 10 11
an
?
3
11
13
15
17
19
21
23
9
7
5
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Check It Out! Example 2b
Find the 11th term of the arithmetic sequence.
9.2, 9.15, 9.1, 9.05,
Step 1 Find the common difference d 9.15
9.2 0.05.
Step 2 Evaluate by using the formula.
an a1 (n 1)d
General rule.
Substitute 9.2 for a1, 11 for n, and 0.05 for
d.
a11 9.2 (11 1)(0.05)
8.7
The 11th term is 8.7.
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Check It Out! Example 2b Continued
Check Continue the sequence.
?
n 1 2 3 4 5 6 7 8 9 10 11
an 9.2 9.15 9.1 9.05 9 8.95 8.9 8.85 8.8 8.75 8.7
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Check It Out! Example 3
Find the missing terms in the arithmetic sequence
Step 1 Find the common difference.
an a1 (n 1)d
General rule.
0 2 (5 1)d
Substitute 0 for an, 2 for a1, and 5 for n.
2 4d
Solve for d.
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Check It Out! Example 3 Continued
1
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Because arithmetic sequences have a common
difference, you can use any two terms to find the
difference.
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Check It Out! Example 4a
Find the 11th term of the arithmetic sequence.
a2 133 and a3 121
Step 1 Find the common difference.
an a1 (n 1)d
a3 a2 (3 2)d
Let an a3 and a1 a2. Replace 1 with 2.
a3 a2 d
Simplify.
121 133 d
Substitute 121 for a3 and 133 for a2.
d 12
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Check It Out! Example 4a Continued
Step 2 Find a1.
an a1 (n 1)d
General rule
Substitute 133 for an, 2 for n, and 12 for d.
133 a1 (2 1)(12)
133 a1 12
Simplify.
145 a1
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Check It Out! Example 4a Continued
Step 3 Write a rule for the sequence, and
evaluate to find a11.
an a1 (n 1)d
General rule.
Substitute 145 for a1 and 12 for d.
a11 145 (n 1)(12)
a11 145 (11 1)(12)
Evaluate for n 11.
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The 11th term is 25.
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Check It Out! Example 4b
Find the 11th term of each arithmetic sequence.
a3 20.5 and a8 13
Step 1 Find the common difference.
an a1 (n 1)d
General rule
Let an a8 and a1 a3. Replace 1 with 3.
a8 a3 (8 3)d
a8 a3 5d
Simplify.
13 20.5 5d
Substitute 13 for a8 and 20.5 for a3.
7.5 5d
Simplify.
1.5 d
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Check It Out! Example 4b Continued
Step 2 Find a1.
an a1 (n 1)d
General rule
Substitute 20.5 for an, 3 for n, and 1.5 for d.
20.5 a1 (3 1)(1.5)
20.5 a1 3
Simplify.
23.5 a1
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Check It Out! Example 4b Continued
Step 3 Write a rule for the sequence, and
evaluate to find a11.
an a1 (n 1)d
General rule
a11 23.5 (n 1)(1.5)
Substitute 23.5 for a1 and 1.5 for d.
Evaluate for n 11.
a11 8.5
The 11th term is 8.5.
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In Lesson 12-2 you wrote and evaluated series. An
arithmetic series is the indicated sum of the
terms of an arithmetic sequence. You can derive a
general formula for the sum of an arithmetic
series by writing the series in forward and
reverse order and adding the results.
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Check It Out! Example 5a
Find the indicated sum for the arithmetic series.
S16 for 12 7 2 (3)
Find the common difference.
d 7 12 5
Find the 16th term.
a16 12 (16 1)(5)
63
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Check It Out! Example 5a Continued
Find S16.
Sum formula.
Substitute.
16(25.5)
Simplify.
408
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Check It Out! Example 5b
Find the indicated sum for the arithmetic series.
Find 1st and 15th terms.
a1 50 20(1) 30
a15 50 20(15) 250
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Check It Out! Example 5b Continued
Find S15.
Sum formula.
Substitute.
15(110)
Simplify.
1650
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Check It Out! Example 6a
What if...? The number of seats in the first row
of a theater has 14 seats. Suppose that each row
after the first had 2 additional seats.
How many seats would be in the 14th row?
Write a general rule using a1 14 and d 2.
an a1 (n 1)d
Explicit rule for nth term
a14 11 (14 1)(2)
Substitute.
11 26
Simplify.
37
There are 37 seats in the 14th row.
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Check It Out! Example 6b
How many seats in total are in the first 14 rows?
Find S14 using the formula for finding the sum of
the first n terms.
Formula for first n terms
Substitute.
Simplify.
There are 336 total seats in rows 1 through 14.