Geometric sequence and Series

GEOMETRIC SEQUENCE &
SERIES
Maribeth Dusal- Alpuerto
Grade 10 Mathematics

Geometric Sequence & Series by Maribeth Dusal- Alpuerto
GEOMETRIC SEQUENCE
• Illustrate a geometric sequence
• Differentiate a geometric from
an arithmetic sequence
• Determines geometric means,
nth term of a geometric
sequence and sum of the terms
of a given geometric sequence
Learning
Competencies

GEOMETRIC SEQUENCE
Warm Up
Find the value of each expression.
1. 25 2. 2–5
3. –34 4. (–3)4
32
5. (0.2)3 6. 7(–4)2
–81 81
1120.008
7. 8. 12(–0.4)3
–0.768

GEOMETRIC SEQUENCE
The table shows the heights of a bungee
jumper’s bounces.
The height of the bounces shown in the table
above form a geometric sequence. In a
geometric sequence, the ratio of successive
terms is the same number r, called the
common ratio.

GEOMETRIC SEQUENCE
Geometric sequences can be thought of as
functions. The term number, or position in the
sequence, is the input, and the term itself is the
output.
a1 a2 a3 a4
1 2 3 4 Position
3 6 12 24 Term
To find a term in a geometric sequence, multiply
the previous term by r.

GEOMETRIC SEQUENCE
The variable a is often used to represent terms in
a sequence. The variable a4 (read “a sub 4”)is the
fourth term in a sequence.
Writing Math

GEOMETRIC SEQUENCE
Example 1A: Extending Geometric Sequences
Find the next three terms in the geometric
sequence.
1, 4, 16, 64,…
Step 1 Find the value of r by dividing each term
by the one before it.
1 4 16 64
The value of r is 4.

GEOMETRIC SEQUENCE
Example 1A Continued
sequence.
1, 4, 16, 64,…
Step 2 Multiply each term by 4 to find the next
three terms.
64 256 1024 4096
4 4 4
The next three terms are 256, 1024, and 4096.

GEOMETRIC SEQUENCE
Example 1B: Extending Geometric Sequences
sequence.
The value
of r is .
–

GEOMETRIC SEQUENCE
When the terms in a geometric
sequence alternate between positive
and negative, the value of r is
negative.
Helpful Hint

GEOMETRIC SEQUENCE
Example 1B Continued
sequence.
Step 2 Multiply each term by to find the next
three terms.
The next three terms are

GEOMETRIC SEQUENCE
Check It Out!
sequence.
5, –10, 20,–40,…
5 –10 20 –40
The value of
r is –2.

GEOMETRIC SEQUENCE
Step 2 Multiply each term by –2 to find the next
three terms.
(–2) (–2) (–2)
The next three terms are 80, –160, and 320.
–40 80 –160 320

GEOMETRIC SEQUENCE
The pattern in
the table shows
that to get the
nth term,
multiply the first
term by the
common ratio
raised to the
power n – 1.
To find the output an of a geometric sequence
when n is a large number, you need an equation,
or function rule.

GEOMETRIC SEQUENCE
If the first term of a geometric sequence is a1,
the nth term is an , and the common ratio is r,
then
an = a1rn–1
nth term 1st term Common ratio

GEOMETRIC SEQUENCE
Example 2A: Finding the nth Term of a Geometric
Sequence
The first term of a geometric sequence is 500,
and the common ratio is 0.2. What is the 7th
term of the sequence?
an = a1rn–1 Write the formula.
a7 = 500(0.2)7–1 Substitute 500 for a1,7 for n, and
0.2 for r.
= 500(0.2)6
Simplify the exponent.
= 0.032 Use a calculator.
The 7th term of the sequence is 0.032.

GEOMETRIC SEQUENCE
Example 2B: Finding the nth Term of a Geometric
Sequence
For a geometric sequence, a1 = 5, and r = 2.
Find the 6th term of the sequence.
an = a1rn–1 Write the formula.
a6 = 5(2)6–1 Substitute 5 for a1,6 for n, and 2
for r.
= 5(2)5
Simplify the exponent.
= 160
The 6th term of the sequence is 160.

GEOMETRIC SEQUENCE
Geometric Mean: The terms between any
two nonconsecutive terms of a geometric
sequence.
Ex. 2, 6, 18, 54, 162
6, 18, 54 are the Geometric Mean between 2
and 162

GEOMETRIC SEQUENCE
Find two geometric means between –2 and 54
-2, ____, ____, 54
1a First term
na nth term
nS sum of n terms
n number of terms
r commonratio
-2
54
4
NA
r
n 1
n 1a a r 

The two geometric means are 6 and -18, since –2, 6, -18, 54
forms a geometric sequence
Example:

GEOMETRIC SERIES

GEOMETRIC SERIES
Recall Vocabulary of Sequences (Universal)
1a First term
na nth term
r commonratio

GEOMETRIC SERIES
Application: Suppose you e-mail a joke to three friends on Monday. Each of those friends sends the joke to three of
their friends on Tuesday. Each person who receives the joke on Tuesday sends it to three more people on Wednesday,
and so on.
Monday
Tuesday
# New people that receive joke Day of Week Total # of people that received joke
Monday
Tuesday
Wednesday
3 3
9 3 + 9 = 12
27 12 + 27 = 39

GEOMETRIC SERIES
1a First term
na nth term
r commonratio
3
10
Sn
NA
Find the sum of the first 10 terms of the
geometric series 3 - 6 + 12 – 24+ …
-2

GEOMETRIC SERIES
In the book Roots, author Alex Haley traced
his family history back many generations to
the time one of his ancestors was brought to
America from Africa. If you could trace your
family back 15 generations, starting with
your parents, how many ancestors would
there be?
1a First term
na nth term
r commonratio
2
15
Sn
NA
2

ARITHMETIC OR GEOMETRIC?
What is the difference
between arithmetic
sequence and geometric
sequence?

Geometric sequence and Series

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