ALGEBRA 2: 12.1 Define and Use Sequences and Series
Goal  Recognize and write rules for number patterns.
VOCABULARY
Sequence
A function whose domain is a set of consecutive integers
Terms
The values in the range of a sequence
Series
The expression that results when the terms of a sequence are added together
Summation notation
Notation for a series that represents the sum of the terms
Sigma notation
Another name for summation notation, which uses the uppercase Greek letter, sigma, written 
SEQUENCES
A sequence is a function whose domain is a set of _consecutive_ integers. If a domain is not specified,it is
understood that the domain starts with 1. The values in the range are called the _terms_ of the sequence.
Domain:1
2
3
4 … n The relative position of each term
Range: a1
a2
a3
a4 … an Terms of the sequence
A _finite_ sequence has a limited number of terms.
An _infinite_ sequence continues without stopping.
Finite sequence: 2, 4, 6, 8
Infinite Sequence: 2, 4, 6, 8, …
A sequence can be specified by an equation, or _rule_ .For example, both sequences above can be described by
the rule an = 2n or f(n) = 2n.
Example 1
Write terms of sequences
Write the first six terms of an = 2n + 1.
a1 = ______ = ____
1st term
a2 = ______ = ____
2nd term
a3 = ______ = ____
3rd term
a4 = ______ = ____
4th term
a5 = ______ = ____
5th term
a6 = ______ = ____
6th term
Example 2
Write rules for sequences
Describe the pattern, write the next term, and write a rule for the nth term of the sequence
(a) 1, 4, 9,16, … and (b) 0, 7, 26, 63, …
Checkpoint Complete the following exercises.
1. Write the first six terms of the sequence f(n) = 3n  7.
2. For the sequence 3, 9, 27, 81, …, describe the pattern, write the next term, and write a rule for the nth
term.
SERIES AND SUMMATION NOTATION
When the terms of a sequence are added together, the resulting expression is a series. A series can be finite or
infinite.
Finite series: 2 + 4 + 6 + 8
Infinite series: 2 + 4 + 6 + 8 …
You can use __summation__ notation to write a series.
4
2  4  6  8   2i
i 1

2  4  6  8     2i
i 1
For both series, the index of summation is __i__ and the lower limit of summation is __1__. The upper limit of
summation is __4__ for the finite series and ____( infinity ) for the infinite series. Summation notation is also
called __sigma__ notation because it uses the uppercase Greek letter sigma, written .
Example 3
Write series using summation notation
Write the series using summation notation.
1 1
1


a. 4 + 7 + 10 + … + 46
b. 1  
8 27 64 
Checkpoint Write the series using summation notation.
3. 7 + 14 + 21 + … + 77
4. 4 8 12 16  …
Example 4
Find the sum of a series
Find the sum of the series.
5
 2  3k
k 3
=
FORMULAS FOR SPECIAL SERIES
Sum of n terms of 1
n
1  n
Sum of first n positive
integers
n
1 
i 1
i 1
Sum of squares of first n
positive integers
nn  1
2
n
i
i 1
Example 5
Use a formula for a sum
32
Use a formula for special series to find the sum of  i.
i 1
32
i =
i 1
Checkpoint Find the sum of the series.
 k
8
k 4
2
- 6
28
i
i 1
2
2

nn  12n  1
6
12.2 Analyze Arithmetic Sequences and Series
Goal  Study arithmetic sequences and series.
VOCABULARY
Arithmetic sequence
A sequence in which the difference between consecutive terms is constant
Common difference
The constant difference between terms of an arithmetic sequence, denoted by d
Arithmetic series
The expression formed by adding the terms of an arithmetic sequence, denoted by Sn
Example 1
Identify arithmetic sequences
Tell whether the sequence 5, 3, 1, 1, 3,... is arithmetic.
.
Checkpoint Decide whether the sequence is arithmetic.
1. 32, 27, 21, 17, 10, . . .
RULE FOR AN ARITHMETIC SEQUENCE
The nth term of an arithmetic sequence with first term a1 and common difference d is given by:
an  a1  (n  1)d
Example 2
Write a rule for the nth term
Write a rule for the nth term of the sequence. Then find a19.
a. 2, 9, 16, 23, . . .
b. 57, 45, 33, 21, . . .
Checkpoint Write a rule for the nth term of the arithmetic sequence. Then find a22.
2. 9, 5, 1, 3, . . .
3. 15, 9, 3, 3, . . .
Example 3
Write a rule given a term and common difference
One term of an arithmetic sequence is a11  41. The common difference is d  5.
(a) Write a rule for the nth term.
Example 4
Write a rule given two terms
Two terms of the arithmetic sequence are a6  7 and a22  87. Find a rule for the nth term.
Checkpoint Write a rule for the nth term of the arithmetic sequence. Then find a22
1.
a15  107, d  12
2.
a5  91, a20  1
THE SUM OF A FINITE ARITHMETIC SERIES
The sum of the first n terms of an arithmetic series is:
 a  an 
Sn  n  1

 2 
In words, Sn is the _mean_ of the _first and nth_ terms, _multiplied_ by
_the number of terms_.
Example 5
Find a sum
15
Find the sum of the arithmetic series
 (9  3i).
i 1
Checkpoint Find the sum of the arithmetic series.
18
3.
 (77  4i)
i 1
12.3 Analyze Geometric Sequences and Series
Goal • Study geometric sequences and series.
VOCABULARY
Geometric sequence
A sequence in which the ratio of any term to the previous term is constant
Common ratio
The constant ratio between consecutive terms of a geometric sequence, denoted by r
Geometric series
The expression formed by adding the terms of a geometric sequence
Example 1
Identify geometric sequences.
Tell whether the sequence 1,4,16, 64, 256,… is geometric.
Checkpoint Tell whether the sequence is geometric.
1. 512, 128, 64, 8,…
RULE FOR A GEOMETRIC SEQUENCE
The nth term of a geometric sequence with first term a1 and common ratio r is given by:
an = a1rn1
Example 2
Write a rule for the nth term
Write a rule for the nth term of the sequence 972, 324,108, 36, …. Then find a10
Example 3
Write a rule given a term and common ratio
One term of a geometric sequence is a3 = 18. The common ratio is r = 3. (a) Write a rule for the nth
term.
Checkpoint Write a rule for the nth term of the geometric sequence. Then find a9.
2. 14, 28, 56, 112,…
3. a5 = 324, r = 3
Example 4
Write a rule given two terms
Two terms of a geometric sequence are a2 = 10 and a7 = 320. Find a rule for the nth term.
Checkpoint Write a rule for the nth term of the geometric sequence. Then find a9.
1.
a3 = 224, a6 = 28
THE SUM OF A FINITE GEOMETRIC SERIES
The sum of the first n terms of a geometric series with common ratio r  1 is:
 1 r n 


s n a1
 1 r 
Example 5
Find the sum of a geometric series
13
Find the sum of the geometric series  34  .
i 1
i 1
Checkpoint Find the sum of the geometric series.
11
2.  7 5n 1
i 1
12.4 Find Sums of Infinite Geometric Series
Goal  Find the sums of infinite geometric series.
VOCABULARY
Partial sum
The sum Sn of the first n terms of an infinite series
THE SUM OF AN INFINITE GEOMETRIC SERIES
The sum of an infinite geometric series with first term a1 and common ratio r is given by
S=
a1
1 r
provided r < 1. If r  1, the series has _no sum_ .
Example 1
Find sums of infinite geometric series
Find the sum of the infinite geometric series.

a.
 6(0.6)i 1

i1
2 4 8
b. 1     ...
3 9 27
c. 1  2 + 4  8 + …
Checkpoint Find the sum of the infinite geometric series, if it exists.
 9  k
5 

k 1  7 

1.
1
 5  n
2.  9 
n 1  6 

3. 6 
1
10 50 250


 . ..
3
27 243
Example 3
Write a repeating decimal as a fraction
Write 0.474747 . . . as a fraction in lowest terms.
Checkpoint Write the repeating decimal as a fraction.
4. 0.888 . . .
5.
0.636363 . . .
12.5 Use Recursive Rules with Sequences and Functions
Goal  Use recursive rules for sequences.
Your Notes
VOCABULARY
Explicit rule
A rule for a sequence that gives an as a function of the term's position number n
Recursive rule
A rule for a sequence that gives the beginning term or terms of a sequence and then a recursive equation that
tells how an is related to one or more preceding terms
Iteration
The repeated composition of a function f with itself
Example 1
Evaluate recursive rules
Write the first six terms of the sequence.
a0 = 2, an = an  1 3
Checkpoint Write the first five terms of the sequence.
1. a0 = 4, an = 1.5an1
RECURSIVE EQUATIONS FOR ARITHMETIC AND GEOMETRIC SEQUENCES
Arithmetic Sequence an = an1+ d where d is the common difference
Geometric Sequence an = r  an1 where r is the common ratio
Example 2
Write recursive rules
Write a recursive rule for the sequence.
a. 1, 7, 13, 19, 25, . . .
b. 4, 12, 36, 108, 324, . . .
Example 3
Write recursive rules for special sequences
Write a recursive rule for the sequence 3, 5, 2, 3, 5,. . ..
Checkpoint Write a recursive rule for the sequence.
2. 3.27, 243 2187 19,683,…
3. 89,78, 67, 56, 45,…
4. 9, 4, 13, 17, 30,…