7.9 Geometic Sequences, Part 1: Recursive Equations
Act 1: Definition of a Geometric Sequence
We have studied arithmetic sequences in which there was a common difference between consecutive
terms that was constant.
1, 3, 5, 7, …
a.
The sequence above is arithmetic. How do you know that?
A geometric sequence is a sequence where the common ratio between terms is constant.
b. List the terms of this geometric sequence. ____, ____, ____
c. What operation are you using to generate this pattern? __________________________
d. A geometric sequence is a sequence where the common ratio between terms is constant.
What do we mean by common ratio?
e. What is the common ratio in each of these geometric sequences?
3, 12, 48, 192, …
56, 28, 14, 7, …
-500, -100, -20, -4, …
common ratio: ___
common ratio: ___
common ratio: ___
f. Generate the next three terms in each sequence.
12, 6, 3, …
____, ____, ____
20, 40, 80, …
____, ____, ____
-3, 15, -75, 375, …
____, ____, ____
g. List each of the following sequences as arithmetic, geometric, or neither. Explain how you know.
Arithmetic, Geometric, How do you know? Explain.
Sequence
or Neither
2, 4, 8, 16, ….
2, -6, 18, -54, ….
3, 15, 45, 225, …
200, 100, 50, 25, …
4, 8, 12, 16, …
5
80, 20, 5, , …
4
5, 6, 8, 11, …
1
Act 2: Writing Equations for Geometric Sequences
Recall that you could write equations for arithmetic sequences in either recursive or explicit form.
Recursive equation:
An = An-1 + 2,
1
3
5
7
A1 = 1
Explicit equation:
y = 2(x – 1) + 1 or f(x) = 2(x-1) + 1
a. We usually use the explicit equation to find the value of further terms. How many circles would
be in the 80th term?
Let’s see how similar it is to write equations for geometric sequences. Fill in the blanks for the sequence.
b. Recursive equations:
An = An-1 __ ___ ,
A1 = ____
1
3
9
c. Geometric sequences are not linear functions. What kind of function are they?
d. Write an explicit equation for the sequence.
y= _______________________
e. Write two equations for each of the geometric sequences.
Recursive Equations
Sequence
“an “ equations
Explicit Equation
“y=” or “f(x)=”
5, 30, 180, 1080, …
1000, 200, 40, 8, …
Act 3: Practice
a. What are the next 5 terms in the geometric sequence? .25, .75, 2.25, ….
____, ____, ____, ____, ____
b. Determine if each sequence is arithmetic, geometric, or neither and justify your answer. Write
equations for arithmetic or geometric sequences.
Arithmetic,
Recursive Equation
Justify.
Explicit Equation
Geometric,
in
Subscript
Notation
Sequence
“y=”
or Neither
“an”
3, 9, 15, 21 …
27, 9, 3, 1 …
-3, 9, -27 …
3
4
1
1
2
4
, , ,0, . . .
c. Is this an arithmetic or geometric sequence? 4, 8, …
d. How many terms do you need to know in order to determine whether a list of numbers is an
arithmetic or geometric sequence? Be ready to explain.
2