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Unit 2
Lesson 20
Geometric Sequences
Objectives
 I can decipher a pattern in a number sequence
 I can write explicit and recursive formulas of geometric sequences
Warm Up: write the next 3 terms of the sequence
2, 2, 4, 6, 10,…..
1, 3, 9, 27, 81,……
60, 30, 15, 7.5,…..
5, 25, 125, 625,……
A sequence is an arrangement of numbers or objects that follow
a rule or a pattern.
Each number in a sequence is called a term. The variable a is
often used to stand for the terms in a sequence. The first term
is 𝒂𝟏 , the second is 𝒂𝟐 , and so on. The nth term of the sequence
is written 𝒂𝒏 , where n can be any positive integer.
In a geometric sequence, each term is found by multiplying the
previous term by a fixed number, called the common ratio (r).
Are any of the warm ups examples of geometric sequences?
If so, what is r?
There are two ways to write sequence formulas—recursive and
explicit.
In the recursive process, finding a term depends on knowing a
previous term, so you MUST define at least one previous term.
The recursive definition looks like this
The explicit definition looks like this
𝒂𝒏 = 𝒂𝒏−𝟏
∙ r
𝒂𝒏 = 𝒂𝟏 ∙ 𝒓𝒏−𝟏
Let’s Try….
2, 4, 16, 256,…….
Find the next 3 terms in the sequence
Write a recursive equation for the sequence
Write the explicit equation for this geometric sequence
Find the 56th term
A geometric sequence has an initial value of 1024, and each term
in the sequence is half of the previous term. Write an explicit
formula to find any term in the sequence.
Find the 9th term of the sequence.
You try……….
5, 10, 15, 20, 25,…..
write the recursive equation
write the explicit formula
find the 15th term
3, -9, 27, -81,…..
write the recursive equation
write the explicit formula
find the 18th term
A petri dish contains 4 viruses. Each hour, the number of viruses
increases, as shown in the table. The population change can be
modeled by a geometric sequence. Write a recursive formula and an
explicit formula that can model this sequence. Use the formulas to
predict how many viruses will be in the dish by the 11th hour.
Hour (n)
1
2
3
4
5
Population (𝒂𝒏 )
4
12
36
108
324
Name: _________________________
Unit 2 Lesson 20
Determine if the sequence is geometric. If it is, identify the
common ratio.
1.
7, -14, 28, -56, 112, …..
2.
1, 5, 7, 9, …..
3.
0.05, 0.5, 5, 50,…
4.
3, 2, 4/3, 8/9,…..
Write a recursive and an explicit formula for each sequence.
5.
12, 24, 48, 96, …..
6.
¼, 1/16, 1/64, 1/256,…..
𝑎𝑛 = _________________
𝑎𝑛 = _________________
𝑎𝑛 = _________________
𝑎𝑛 = _________________
7. 𝑎1 = 3
8.
r = 20
𝑎1 = 5000
r = 0.2
𝑎𝑛 = _________________
𝑎𝑛 = _________________
𝑎𝑛 = _________________
𝑎𝑛 = _________________
Find the indicated term in the geometric sequence.
9.
If
𝑎1 = 1 and 𝑎2 = 6
10.
If 𝑎1 = 2 and r = -4
find 𝑎7
find 𝑎10
11.
Jenny started a chain letter by e-mail. She sent it to five of her
friends and asked them to each send it to five of their friends.
Assume that no one breaks the chain and that no person receives the
e-mail twice. How many e-mails will be sent during the 6th
generation? (Treat Jenny’s e-mails as the 1st generation of the
letter.)
Name: _________________________
Arithmetic/Geometric Mix HW
Determine if the sequence is arithmetic, geometric, or neither. If
it is arithmetic or geometric, identify the common difference or
common ratio.
1. 5, 9, 13, 17, 21,…..
2. 80, 40, 20, 10,……
3.
18, 14, 10, 6,…….
5. The formula 𝑎𝑛 = 10 – 4n
describes an arithmetic
sequence. What are the first 4
terms in the sequence?
4.
1, 2, 3, 5, 8,…..
6. Write an explicit formula to
find the nth term in the sequence
below.
128, 96, 72, 54, …….
Use the geometric sequence below for questions 7 – 9.
80, 40, 20, 10, …..
7. Write an explicit formula in terms of n to show how to find the nth
term of this sequence.
8. Plot points (n, 𝑎𝑛 ) on the grid to represent the first 5 terms in the
sequence. Find the average rate of change between each adjacent
pair of points.
9. Think of this sequence as a function. What type of function is it?
What are its domain and range?