Algebra I
Geometric Sequences
Name: __________________________________
Date: ______________________
Warm Up
Consider the following sequence explicitly defined below:
𝑎𝑛 = 2𝑛
Find the first four terms in the sequence and graph the sequence below.
Last week, we saw that arithmetic sequences are really just like linear functions whose
domains are {1, 2, 3, 4, …}. Today, we will be exploring sequences that are another type
of functions that we already know!
Geometric Sequences
A geometric sequence is a sequence such that there exists a common ratio between
each of the terms.
We use the variable r to denote the common ratio.
In simpler terms, a sequence is geometric if we multiplying by the same amount
between every term, which remind us of __________________________ functions!
Algebra I
Geometric Sequences
1. Consider the geometric sequence whose first three terms are given by: 6, 18, 54
a. State the common ratio of the sequence (r)
b. Determine the fourth, fifth, and sixth terms of the sequence.
c. Write an explicit formula for 𝑎𝑛 .
d. Find 𝑎15 .
2. For each of the following geometric sequences:
a. Find the common ratio.
b. Fill in the missing term.
c. Write an explicit formula for the nth term.
𝑎𝑛 : 5, 10, 20, ___________ , 80, 160
𝑎𝑛 : 4, -20, 100, __________, 2500
1
1
𝑏𝑛 : 120, 60, 30, ___________, 7.5
𝑐𝑛 : 64, 16, 4 __________, 4 , 16
𝑑𝑛 : 5, -5, 5 _______, 5
𝑒𝑛 : 3, 12, 48, ___________, 768
Algebra I
Geometric Sequences
3. Consider the geometric sequence whose first three terms are 16, 8, 4.
a. Determine the common ratio.
b. Find the next three terms of the sequence and graph the first six terms on the graph
below.
c. Write an explicit formula for 𝑎𝑛 .
4. Consider the following table representing a geometric sequence.
n
1
2
3
f(n)
81
54
36
Determine the common ratio for the sequence.
Finish the table by finding the next two terms.
Write an explicit formula for f(n).
4
5
Algebra I
Geometric Sequences
Name: __________________________________
Date: _________________
Geometric Sequence Homework
1. For each of the following geometric sequences:
i. State the common ratio.
ii. Write an explicit formula for the nth term in the sequence.
a.) 15, 45, 135, 405, ….
b.) 8, 20, 50, …
c.)
n
f(n)
1
120
2
180
3
270
4
405
Algebra I
Geometric Sequences
2. Write a formula for 𝑎𝑛 , the nth term, of the geometric sequence given that 𝑎1 = 20
and 𝑎2 = 4.
3. Maria plans to double the amount of time she spends walking per day each week.
She starts, on week 1, walking 4 minutes per day. On week 2, she then walks 8
minutes per day, etcetera.
a. Determine the first six terms of this sequence.
b. Write an explicit rule for a(n), the amount of time Maria walks on week n.
c. Graph the geometric sequence on the graph to the right.