12-3
12-3Arithmetic
ArithmeticSequences
Sequencesand
andSeries
Series
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
12-3 Arithmetic Sequences and Series
Warm Up
Find the 5th term of each sequence.
1. an = n + 6 11
2. an = 4 – n –1
3. an = 3n + 4 19
Write a possible explicit rule for the nth term
of each sequence.
an = n + 3
4. 4, 5, 6, 7, 8,…
6.
Holt Algebra 2
an = 2n – 5
5. –3, –1, 1, 3, 5, …
12-3 Arithmetic Sequences and Series
Objectives
Find the indicated terms of an
arithmetic sequence.
Find the sums of arithmetic series.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Vocabulary
arithmetic sequence
arithmetic series
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 1A: Identifying Arithmetic Sequences
Determine whether the sequence could be
arithmetic. If so, find the common first
difference and the next term.
–10, –4, 2, 8, 14, …
–10,
Differences
–4,
6
2,
6
8,
6
14
6
The sequence could be arithmetic with a common
difference of 6.
The next term is 14 + 6 = 20.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 1B: Identifying Arithmetic Sequences
Determine whether the sequence could be
arithmetic. If so, find the common first
difference and the next term.
–2, –5, –11, –20, –32, …
–2,
Differences
–5,
–3
–11,
–6
–20,
–32
–9
–12
The sequence is not arithmetic because the first
differences are not common.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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,
Differences
1.2,
–0.7
0.5,
–0.7
–0.2,
–0.7
–0.9
–0.7
The sequence could be arithmetic with a common
difference of –0.7.
The next term would be –0.9 – 0.7 = –1.6.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 2: Finding the nth Term Given an Arithmetic
Sequence
Find the 12th term of the arithmetic sequence
20, 14, 8, 2, 4, ....
Step 1 Find the common difference: d = 14 – 20 = –6.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 2 Continued
Step 2 Evaluate by using the formula.
an = a1 + (n – 1)d
a12
General rule.
= 20 + (12 – 1)(–6) Substitute 20 for a1, 12 for n, and
–6 for d.
= –46
The 12th term is –46.
Check Continue the sequence.
Holt Algebra 2

12-3 Arithmetic Sequences and Series
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.
a11= –3 + (11 – 1)(–2)
= –23
The 11th term is –23.
Holt Algebra 2
Substitute –3 for a1, 11 for n,
and –2 for d.
12-3 Arithmetic Sequences and Series
Check It Out! Example 2a Continued
Check Continue the sequence.
n
1
2
3
4
5
6
7
8
9
10 11
an –3 –5 –7 –9 –11 –13 –15 –17 –19 –21 –23 
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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.
a11= 9.2 + (11 – 1)(–0.05)
Substitute 9.2 for a1, 11 for n,
and –0.05 for d.
= 8.7
The 11th term is 8.7.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Check It Out! Example 2b Continued
Check Continue the sequence.
n
1
an
9.2
Holt Algebra 2
2
3
4
9.15 9.1 9.05

5
6
7
8
9
10
11
9
8.95
8.9
8.85
8.8
8.75
8.7
12-3 Arithmetic Sequences and Series
Example 3: Finding Missing Terms
Find the missing terms in the arithmetic
sequence
17,
,
,
, –7.
Step 1 Find the common difference.
an = a1 + (n – 1)d
General rule.
–7 = 17 + (5 – 1)(d)
Substitute –7 for an,
17 for a1, and 5 for n.
–6 = d
Solve for d.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 3 Continued
Step 2 Find the missing terms using d= –6 and a1 = 17.
a2 = 17 + (2 – 1)(–6)
= 11
a3 = 17 +(3 – 1)(–6)
=5
a4 = 17 + (4 – 1)(–6)
= –1
Holt Algebra 2
The missing terms are
11, 5, and –1.
12-3 Arithmetic Sequences and Series
Check It Out! Example 3
Find the missing terms in the arithmetic sequence
2,
,
,
, 0.
Step 1 Find the common difference.
an = a1 + (n – 1)d
0 = 2 + (5 – 1)d
–2 = 4d
General rule.
Substitute 0 for an,
2 for a1, and 5 for n.
Solve for d.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Check It Out! Example 3 Continued
Step 2 Find the missing terms using d=
and a1= 2.
The missing terms are
=1
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Because arithmetic sequences have a common
difference, you can use any two terms to find the
difference.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 4: Finding the nth Term Given Two Terms
Find the 5th term of the arithmetic sequence
with a8 = 85 and a14 = 157.
Step 1 Find the common difference.
an = a1 + (n – 1)d
a14 = a8 + (14 – 8)d
Let an = a14 and a1 = a8. Replace
1 with 8.
a14 = a8 + 6d
Simplify.
157 = 85 + 6d
72 = 6d
12 = d
Holt Algebra 2
Substitute 157 for a14 and 85
for a8.
12-3 Arithmetic Sequences and Series
Example 4 Continued
Step 2 Find a1.
an = a1 + (n – 1)d
General rule
85 = a1 + (8 - 1)(12)
Substitute 85 for a8,
8 for n, and 12 for d.
85 = a1 + 84
Simplify.
1 = a1
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 4 Continued
Step 3 Write a rule for the sequence, and evaluate to
find a5.
an = a1 + (n – 1)d
General rule.
an = 1 + (n – 1)(12)
Substitute 1 for a1 and 12 for d.
a5 = 1 + (5 – 1)(12)
Evaluate for n = 5.
= 49
The 5th term is 49.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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
–121 = –133 + d
d = 12
Holt Algebra 2
Simplify.
Substitute –121 for a3 and –133 for a2.
12-3 Arithmetic Sequences and Series
Check It Out! Example 4a Continued
Step 2 Find a1.
an = a1 + (n – 1)d
General rule
Substitute –133 for an, 2 for n,
–133 = a1 + (2 – 1)(12)
and 12 for d.
–133 = a1 + 12
–145 = a1
Holt Algebra 2
Simplify.
12-3 Arithmetic Sequences and Series
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.
a11 = –145 + (n – 1)(12)
Substitute –145 for a1 and 12
for d.
a11 = –145 + (11 – 1)(12)
Evaluate for n = 11.
= –25
The 11th term is –25.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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
a8 = a3 + (8 – 3)d
General rule
Let an = a8 and a1 = a3.
Replace 1 with 3.
a8 = a3 + 5d
Simplify.
13 = 20.5 + 5d
Substitute 13 for a8 and 20.5 for a3.
–7.5 = 5d
–1.5 = d
Holt Algebra 2
Simplify.
12-3 Arithmetic Sequences and Series
Check It Out! Example 4b Continued
Step 2 Find a1.
an = a1 + (n – 1)d
General rule
20.5 = a1 + (3 – 1)(–1.5)
Substitute 20.5 for an, 3 for n,
and –1.5 for d.
20.5 = a1 – 3
Simplify.
23.5 = a1
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Check It Out! Example 4b Continued
Step 3 Write a rule for the sequence, and evaluate to
find a11.
an = a1 + (n – 1)d
a11 = 23.5 + (n – 1)(–1.5)
a11 = 23.5 + (11 – 1)(–1.5)
a11 = 8.5
The 11th term is 8.5.
Holt Algebra 2
General rule
Substitute 23.5 for a1
and –1.5 for d.
Evaluate for n = 11.
12-3 Arithmetic Sequences and Series
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.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 5A: Finding the Sum of an Arithmetic Series
Find the indicated sum for the arithmetic series.
S18 for 13 + 2 + (–9) + (–20) + ...
Find the common difference.
d = 2 – 13 = –11
Find the 18th term.
a18 = 13 + (18 – 1)(–11)
= –174
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 5A Continued
Sum formula
Substitute.
= 18(-80.5) = –1449
Check Use a graphing
calculator.
Holt Algebra 2

12-3 Arithmetic Sequences and Series
Remember!
These sums are actually partial sums. You cannot
find the complete sum of an infinite arithmetic
series because the term values increase or
decrease indefinitely.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 5B: Finding the Sum of an Arithmetic Series
Find the indicated sum for the arithmetic series.
Find S15.
Find 1st and 15th terms.
a1 = 5 + 2(1) = 7
a15 = 5 + 2(15) = 35
= 15(21) = 315
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 5B Continued
Check Use a graphing calculator.

Holt Algebra 2
12-3 Arithmetic Sequences and Series
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
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Check It Out! Example 5a Continued
Find S16.
Sum formula.
Substitute.
= 16(–25.5)
= –408
Holt Algebra 2
Simplify.
12-3 Arithmetic Sequences and Series
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
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Check It Out! Example 5b Continued
Find S15.
Sum formula.
Substitute.
= 15(–110)
= –1650
Holt Algebra 2
Simplify.
12-3 Arithmetic Sequences and Series
Example 6A: Theater Application
The center section of a concert hall has 15
seats in the first row and 2 additional seats in
each subsequent row.
How many seats are in the 20th row?
Write a general rule using a1 = 15 and d = 2.
an = a1 + (n – 1)d
a20 = 15 + (20 – 1)(2)
Explicit rule for nth term
Substitute.
= 15 + 38
Simplify.
= 53
There are 53 seats in the 20th row.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Example 6B: Theater Application
How many seats in total are in the first 20 rows?
Find S20 using the formula for finding the sum of the
first n terms.
Formula for first n terms
Substitute.
Simplify.
There are 680 seats in rows 1 through 20.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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
a14 = 11 + (14 – 1)(2)
= 11 + 26
= 37
Explicit rule for nth term
Substitute.
Simplify.
There are 37 seats in the 14th row.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
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.
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Lesson Quiz: Part I
1. Determine whether the sequence could be
arithmetic. If so, find the first difference and the
next term. –1, –4, –7, –10, –13, … yes; –3,–16
2. Find the 10th term of the arithmetic sequence
–2, –5, –8, –11, –14, … –29
3. Find the missing terms in the arithmetic
sequence 15, , , , 17. 15.5, 16, 16.5
4. Find the 6th term of the arithmetic sequence with
a9 = 64 and a12 = 88. 40
Holt Algebra 2
12-3 Arithmetic Sequences and Series
Lesson Quiz: Part II
5. Find the indicated sum for
–132
6. The side section of an auditorium has 12 seats
in the first row and 3 additional seats in each
subsequent row. How many seats are in the
10th row? How many seats in total are in the
first 10 rows?
39; 255
Holt Algebra 2