Lesson Title: Identifying and Analyzing Sequences. [This lesson can
be done with any 8th-10th grade math class.]
Summary: This is an inquiry-based approach to learning the different
kinds of sequences, interpreting patterns via inductive
reasoning, and finding the nth term. This topic would
generally take longer than a day or two to cover, this lesson
is merely an introduction.
Key terms: Inductive reasoning, conjecture, arithmetic, geometric,
difference, ratio, recursive, (explicit, implicit), use of
notation, and nth term formulas.
Prerequisite knowledge: Recognizing patterns and being able to
determine the next or missing number from the
pattern(forming conjectures).
NCTM & Ohio Standards Addressed:
-Patterns
-Functions
-Algebra Standard: 5-7 Benchmark A, B 8-10 Benchmark A
-Data Analysis and Probability Standard: 5-7 Benchmark B
Learning Objectives: As a result of this lesson, students will be able
to determine if a sequence is arithmetic,
geometric, or neither. They will also predict
specific terms and even come up with their own
sequences.
Materials: Pencil, paper. (Calculator is optional.)
Procedure: Make sure students have paper and pencils out and are ready
to start. Instructor will lead students down the
"sequences" path by demonstrating on the board various
polygons and the total degrees of their interior angles.
The instructor will keep the students active through
questioning ("What kind of polygon is this – draw a
triangle, then this -- draw a quadrilateral, etc.) Next, we
will examine how many total degrees make up the interior
angles of each polygon. Most importantly, the instructor
will let the students discover for themselves the
relationship and pattern that is beginning to develop.
Assignment to follow (Activity) in class and for HW
Investigation (Extension problems)
Assessments: Instructor will assess students orally during the lesson
to determine their level of understanding. The in class
and homework assignments will be collected for further
analysis of to what degree the students are getting it.
Project AMP Dr. Antonio R. Quesada Director, Project AMP
As you go through your everyday life you encounter patterns of every type. Some of these patterns are more noticeable like the rising and setting of the sun. However, some patterns are little less obvious like the rpm’s of a running car engine. In mathematics we encounter many numerical patterns. These patterns can be often be used to create or even predict future values of a particular pattern function. Let’s examine one of these mathematical patterns…
Consider the Polygons. Work with a partner. Record your data in the table below.
-Write the number of sides.
-Divide each polygon into non-overlapping triangles.
-To find the Sum of the Interior Angles multiply the # of triangles formed by 180.
Polygon
/ # of sides / # of triangles formed / Sum of the Interior Angles.Now let’s examine more closely the column labeled, Sum of the Interior Angles.
Do you notice a pattern? What is it?
What would be the Sum of the Interior Angles for a polygon with 7 sides? 9 sides? 10 sides?
Patterns like these are known as arithmetic sequences. Later we will define two specific types of sequences, arithmetic and geometric. We will eventually explore how the model these patterns with general formulas and use these formulas to find any term of a given sequence.
Arithmetic and Geometric Sequences
This lesson is designed to help the student understand the concepts of arithmetic and geometric sequences. The student should be able to identify sequences, generate sequences, find the nth term, and apply real-life applications of sequences.
Definition A sequence is an ordered list of numbers that are related to each other by a pattern. The numbers in the sequence are called the terms.
1) In each sequence below, find the next three terms and describe the pattern.
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
_____ , _____ , ______
Now we will place each of the sequences from #1 into one of the following categories: the same number was repeatedly added, the same number was repeatedly multiplied, or neither of these is true. Recall that a number that remains unchanged is called a constant.
2) List each of the sequences from #1 in one of the columns below:
Adding a Constant Multiplying a Constant Neither of These
Part 1.
One way to examine a sequence is to compare consecutive terms in the sequence.
1) Consider the sequence {6, 10, 14, 18, 22, …}.
a) Describe how you can find a term in the sequence, if you know the preceding
term.
b) How do you find the 2nd term, if you know the 1st term? How do you find the 3rd
term, if know the 1st term? How do you find the 4th term if you know the 1st
term? Hint: if you are having trouble, make a table.
c) How do you find the 10th term without finding the preceding 9 terms? What is the
10th term?
d) How do you find the 100th term without finding the first 99 terms?
e) Write an algebraic expression for the nth term in the sequence.
2) Now consider the sequence {15, 10, 5, 0, -5, …}.
a) How does the sequence change from one term to the next?
b) How do you find the 2nd term if you know the 1st term? How do you find the 3rd
term if you know the 1st term? How do you find the 5th term if you know the 1st
term? What are these terms?
c) How do you find the 20th term without finding the preceding 19 terms? What is
the 20th term?
d) What is the 100th term?
e) Find an algebraic expression for the nth term.
3) Arithmetic sequences are sequences where the terms increase or decrease by the
same amount every time. That is, if you subtract any two consecutive terms, you will
the same number every time. Let d, the difference between consecutive terms, be
defined by d = an – an-1. Write an expression for the nth term, an, in terms of d, n, and
a1, the first term.
Part 2
4) Now look at the sequence {5, 10, 20, 40, 80, 160, …}.
a) How do you find a term in this sequence if you know the term that comes before
it? Is the sequence arithmetic?
b) How do you find the 2nd term if you know the 1st term? How do you find the 3rd
term if you know the 1st term? How do you find the 4th term if you know the 1st
term
c) How do you find the 10th term without find the 1st 9 terms? What is the 10th term?
d) How do you find the 20th term without finding the all terms that precede it? What
is the 20th term?
e) Write an algebraic expression to find the nth term of the sequence.
5) Consider the sequence {800, -200, 50, -12.5, 3.125, …}.
a) What is happening between consecutive terms? How do you find a term if you
know the preceding term?
b) How do you find the 2nd term if you know the 1st term? How do you find the 3rd
term? How do you find the 6th term if you know the 1st term/
c) How do you find the 10th term without finding the preceding terms? What is the
10th term?
d) What is the 15th term?
e) Write an algebraic expression to find the nth term of the sequence.
6) A geometric sequence is a sequence where the ratio of consecutive terms is a
constant. Let r be the ratio between consecutive terms, where r = an_ .
an-1
So r is what you multiply a term by to get the next term in the sequence. Write
An algebraic expression for an, the nth term of a geometric sequence, in terms of
n, r, and a1.
Name ______Date ______
Arithmetic and Geometric Sequences
1. The Math Department at an Akron area high school, as a fundraiser, purchased a candy vending machine for the cafeteria. The department pays a monthly service charge of $35 plus $5 per case of candy to a local distributor to service the machine.
a. The monthly service charge depends on the number of cases sold. Is the sequence of monthly charges ( for 0 cases sold, 1 case sold, 2 cases sold, and so on) arithmetic, geometric, or neither? Explain. (Make a table if necessary)
b. Determine a formula to find the fee for n number of cases sold.
c. What is the monthly fee if 20 cases of soft drink are sold? Explain or
show your work.
2. While cleaning the garage you are cut by a rusty nail that puts 25 bacteria cells
into a wound and then the number of bacteria triples every 15 minutes.
a. The number of bacteria depends on the number of minutes the bacteria
remains in the wound without medical attention. Is the sequence of the
number of bacteria present ( for 15 minutes, 30 minutes, 45 minutes and
so on) arithmetic, geometric or neither? Explain. (Make a table if necessary)
b. Determine a formula to find the number of bacteria after n minutes.
c. How many bacteria would be present if you did not clean the wound for 2 hours?
Project AMP Dr. Antonio R. Quesada Director, Project AMP