Isla Porras
Lesson Plan Title: Geometric Sequence
Standards Addressed: Algebra II - TEKS
(11) Exponential and logarithmic functions. The student formulates equations and inequalities
based on exponential and logarithmic functions, uses a variety of methods to solve them, and
analyzes the solutions in terms of the situation. The student is expected to:
General Goal(s): (F) analyze a situation modeled by an exponential function, formulate an
equation or inequality, and solve the problem.
Specific Objectives: Students will be able to formulate a general equation for a geometric
sequence and they will be able to utilize it in real-world situations. Upon formulating the general
equation, students will realize that geometric sequences are exponential in nature.
Required Materials: Worksheet
Anticipatory Set (Lead-In): “How many of you would like to receive an ‘A’ every day for
a month in my class? What if I were to tell you that I would give you an ‘A’ if you would give me
$0.01, but the price would go up geometrically. For example, on the second day of the month
you’d give me $0.02, the third day you would give me $0.04, and so on and so forth. How many
of you would accept that bribe? Can anyone figure out how much money you’d give me on the
30th day?”
Step-By-Step Procedures:
1. Define a geometric sequence as “a sequence of numbers where each term after the first
is found by multiplying the previous one by a fixed, non-zero number, called the common
ratio.” (http://en.wikipedia.org/wiki/Geometric_sequence)
2. Introduce to students the derivation of the equation to find a general term within a
geometric sequence:
In order to find the nth term, denoted as an there must be a common ratio,
denoted by r.
The first term is written as a1.
a1
a2 = a1r
a3 = a2r = a1r2
a4 = a3r = a1r3
Notice that the exponent r is 1 less than the number of the term. Thus, from that
observation, the nth term of the geometric sequence is found and written as:
an = a1rn-1
3. Work out a couple problems together with students, then have them work out problems
on their worksheet. This time can also be used to link what we have learned to the
anticipatory set.
Plan For Independent Practice: Allow students to take the worksheet home and
complete it as homework. Also, present students with the paper folding sequence on pg. 140 of
Teaching Mathematics book and have them answer the related questions. The homework
assignment should only be about 30 minutes long to help students develop confidence in the
material.
- Closure (Reflect Anticipatory Set): “Now that we’re able to figure out how much
money you would have to pay me each day to get an automatic ‘A’ in my class, how many of you
would want to take me up on my offer? How many of you think it’d be easier to just come to
class and do your work to earn an ‘A’?”
- Assessment Based On Objectives: Engage students with various questions that
slightly change the problems provided on their worksheet to ensure students understand the
concepts and how the changes apply to any of their given situations.
- Extensions (For Gifted Students): Ask students if they can figure out the sum of all
the terms we were able to figure out. Then introduce the derivation for the sum of a geometric
sequence:
Derivation of the formula:
Sn = a1 + a1r + a1r2 +… + a1rn-2 + a1rn-1
r Sn = a1r + a1r2 + a1r3 +… + a1rn-1 + a1rn
Sn – r Sn = a1 - a1rn
(1-r) Sn = a1(1 - rn)
Sn = a1 (1 – rn)/ (1- r)
- Possible Connections To Other Subjects: Compare the relationship and behavior of
this function to exponential growth/decay models. Discuss how exponential functions change
compared to linear functions.
Geometric Sequence
1. Find the eighth term of 4, 8, 16, 32, …
2. Write the fourth term after the first term and the common ratio are given:
A) a = 2, r = 3
B) a = 1/2, r = 4
C) a = -27, r = 2/3
3. Each year, the value of a car is 70% of its value of the previous year. At the end of the
first year, the value of the car was $6,000. What was its value at the end of 3 years?
4. If you were given a gift of $1 on your first birthday, and the gift was doubled on each
of the following birthday, how much would you receive on your twenty-first birthday?
5. The value of a certain rare coin increases by one-tenth each year. If the coin is worth
$3.00 now, what will be its approximate value in 5 years?
Sum of Geometric Sequence
1. Find the sum of the first 8 terms of 4, 8, 16, 32, …
2. Suppose that you save $128 in January and that each month thereafter you only
manage to save half of what you saved the previous month. How much do you save in
the tenth month, and what are your total savings after 10 months.
3. In a lottery, the first ticket drawn paid a prize of $30,000. Each succeeding ticket paid
half as much as the preceding one. If six tickets were drawn, what was the total of prize
money paid?
4. The half-life of Uranium 230 isotope is 20.8 days, that is, one-half of a given amount
of Uranium 230 decomposes every 20.8 days. How much of an initial amount of 1000
grams of the isotope will be left after 208 days?
Solutions for the Student Worksheet
Geometric Sequence
1. 512
2. A). 54 B). 32 C). -8
3. 2940
4. $1,048,576
5. $4.83
Sum of Geometric Sequence
1. 1020
2. $255.75
3. $59,062.50
4. 0.98 grams
Sources
http://www.beaconlearningcenter.com/Lessons/2571.htm
http://en.wikipedia.org/wiki/Geometric_sequence
http://www.utdanacenter.org/intensifiedalgebra/structure.php
Teaching Mathematics: A Sourcebook of Aids, Activities, & Strategies, by Evan M.
Maletsky, Max /a. Sobel.
http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.html