Avon High School
Section: 11.3
ACE COLLEGE ALGEBRA II - NOTES
Geometric Sequences and Series
Mr. Record: Room ALC-129
Semester 2 - Day 38
Congratulations! You just landed your first two job interviews and have
discovered that both companies have offered you a position. Company A
offers you a $30,000 salary the first year with a guranteed 6% increase
per year for four years after that. Company B offers an initial annual salary
of $32,000 with annual increases of 3% per year after that. Over a five
year period, which is the better offer?
This is the type of problem we will address in this section.
Geometric Sequences
The figure below shows a sequence in which the number of squares in increasing.
Notice the number of squares grows as follows: 1, 5, 25, 125, and 625.
Definition of a Geometric Sequence
An geometric sequence is a sequence in which each term after the first is obtained by
multiplying the preceding term by a fixed nonzero constant. The amount by which we
multiply each time is called the common ratio of the sequence.
Activity: Find the common ratio for each geometric sequence.
a. 4, 8, 16, 32, 64, …
Example 1
b. -6, 12, 24, -48, -96 …
1 1
c. 9,  3,1,  , ,
3 9
Writing the Terms of a Geometric Sequence
Write the first six terms of the geometric sequence with first term 12 and the
1
common ratio 
2
The General Term of a Geometric Series
If we were to build the first six terms of a geometric sequence whose first term is a1 and whose common ratio is
r, we would have the following:
General Term of a Geometric Sequence
The nth term (the general term) of a geometric sequence with first term a1 and common ratio r is
an  a1r n 1
Example 2
Using the Formula for the General Term of a Geometric Sequence
Find the seventh term of the geometric sequence whose first term is 5 and whose common
ratio is -3.
Example 3
Geometric Population Growth
The table below shows the population of the United States in 2000, with estimates given by the Census Bureau
for 2001 through 2006.
a. Show the population is increasing geometrically.
b. Write the general term for the geometric sequence modeling the population of the United States, in millions,
n years after 1999.
c. Project the U.S. population, in millions, for the year 2014.
The Sum of the First n Terms if a Geometric Sequence
Just like that of an arithmetic sequence, the sum of the first n terms of a geometric sequence, denoted by S n ,
and called the nth partial sum, can be found without having to add up all the terms. Again, this is especially
nice when we have many terms in our sequence.
The Sum of the First n Terms of a Geometric Sequence
The sum, Sn , of the first n terms of a geometric sequence is given by
a (1  r n )
Sn  1
1 r
In which a1 is the first term and r is the common ratio (r ≠ 1).
PROOF:
Example 4
Finding the Sum of the First n Terms of a Geometric Sequence
Find the sum of the first nine terms of the geometric sequence 2, -6, 18, -54, . . . .
Example 5
Using Sn to Evaluate a Summation
8
Find the following sum
 23
i
i1
.
Example 6
Computing a Lifetime Salary
A union contract specifies that each worker will receive a 5% pay increase each year
for the next 30 years. One worker is paid $20,000 the first year. What is the person’s
total lifetime salary over a 30-year period?
Annuities
Value of an Annuity: Interest Compounded n Times per Year
If P is the deposit made at the end of each compounding period for an annuity at r percent annual
interest compounded n times per year, the value, A, of the annuity after t years is
 r nt 
P 1    1
 n 

A 
r
n
Example 7
Determining the Value of an Annuity
At age 25, to save for retirement, you decide to deposit $200 at the end of each month into an IRA that pays
7.5% compounded monthly.
a. How much will you have from the IRA when you retire at age 65?
b. Find the total interest you earned.
Infinite Geometric Series
Suppose we wanted to add the following numbers
1 1 1
  
2 4 8
Notice that there are an infinite number of terms to add. Can this be done?
1
The Sum of an Infinite Geometric Series
If -1 < r < 1 (equivalently |r| < 1), then the sum of the infinite geometric series
a1  a1r  a1r 2  a1r 3 
in which a1 is the first term and r is the common ratio, is given by
a
S 1
1 r
If |r|≥ 1, the infinite series does not have a sum.
Example 8
Finding the Sum of an Infinite Geometric Series
Find the sum of the infinite geometric series:
3 3 3 3
  
a.
8 16 32 64
Example 9
b. 3  2 
Writing a Repeating Decimal as a Fraction
Express each of the following as a fraction in lowest terms.
a. 0.87
b. 0.9
4 8
 
3 9
Example 10
Tax Rebates and the Multiplier Effect
A tax rebate that returns a certain amount of money to taxpayers can have a total effect on
the economy that is many times the amount. In economics, this phenomenon is called the
multiplier effect. Suppose, for example, that the government reduces taxes so that each
consumer has $2000 more income. The government assumes that each person will spend
70% of this (= $1400). The individuals and businesses receiving this $1400 in turn will
spend 70% of it (= $980), creating extra income for other people to spend, and so on.
Determine the total amount spent on consumer goods from the initial $2000 tax rebate.