A geometric sequence is defined by a constant ratio between consecutive terms, known as the common ratio. This document explains how to identify geometric sequences, provides recursive and explicit formulas for them, and offers examples for finding specific terms in such sequences. It also illustrates problems involving calculating terms based on given sequences and their common ratios.

1. Geometric
Geometric
Sequences
Sequences

2. What is a Geometric Sequence?
What is a Geometric Sequence?
 In a geometric sequence, the ratio
In a geometric sequence, the ratio
between consecutive terms is
between consecutive terms is
constant. This ratio is called the
constant. This ratio is called the
common ratio
common ratio.
.
 Unlike in an arithmetic sequence, the
Unlike in an arithmetic sequence, the
difference between consecutive
difference between consecutive
terms varies.
terms varies.
 We look for
We look for multiplication
multiplication to
to
identify geometric sequences.
identify geometric sequences.

3. A sequence is geometric if the ratios of
consecutive terms are the same.
Geometric Sequence
Geometric Sequence
3
2 4
1 2 3
.....
a
a a
r
a a a
   
The number r is the common
ratio
ratio.

4. Ex: Determine if the sequence is geometric.
Ex: Determine if the sequence is geometric.
If so, identify the common ratio
If so, identify the common ratio
 1, -6, 36, -216
1, -6, 36, -216
yes. Common ratio=-6
yes. Common ratio=-6
 2, 4, 6, 8
2, 4, 6, 8
no. No common ratio
no. No common ratio
 3, 15, 75, 375
3, 15, 75, 375
yes. Common ratio=
yes. Common ratio=

5. Important Formulas for
Important Formulas for
Geometric Sequence:
Geometric Sequence:
 Recursive Formula
Recursive Formula  Explicit Formula
Explicit Formula
an = (an – 1 ) r an = a1 * r n-1
Where:
an is the nth term in the
sequence
a1 is the first term
n is the number of the term
r is the common ratio

6. Ex: Write the recursive formula for
Ex: Write the recursive formula for
each sequence
each sequence
First term: a
First term: a1
1 = 7
= 7
Common ratio = 1/3
Common ratio = 1/3
Recursive: an = an-1 * r
Now find the first five
terms:
a1 = 7
a2 = 7(1/3) = 7/3
a3 = 7/3(1/3) = 7/9
a4 = 7/9(1/3) = 7/27
a5 = 7/27(1/3) = 7/81
an = an-1 * (1/3)

7. Ex: Write the explicit formula for
Ex: Write the explicit formula for
each sequence
each sequence
First term: a
First term: a1
1 = 7
= 7
Common ratio = 1/3
Common ratio = 1/3
Explicit: an = a1 * r n-1
Now find the first five
terms:
a1 = 7(1/3) (1-1)
= 7
a2 = 7(1/3) (2-1)
= 7/3
a3 = 7(1/3) (3-1)
= 7/9
a4 = 7(1/3) (4-1)
= 7/27
a5 = 7(1/3) (5-1)
= 7/81

8. Recursive Geometic Sequence Problem
Recursive Geometic Sequence Problem
Find the 5
Find the 5th
th
and 6
and 6th
th
term in the
term in the
sequence of 11,33,99,297 . . .
sequence of 11,33,99,297 . . .
a6 = 891(3) = 2673
Common ratio = 3
a5 = 297 (3) = 891
Start with the recursive sequence
formula
Find the common ratio
between the values.
Plug in known values
Simplify
an = an-1 * r

9. Explicit Geometic Sequence Problem
Explicit Geometic Sequence Problem
Find the 19
Find the 19th
th
term in the sequence of
term in the sequence of
11,33,99,297 . . .
11,33,99,297 . . .
a19 = 11(3)18
=4,261,626,379
Common ratio = 3
a19 = 11 (3) (19-1)
Start with the explicit sequence formula
Find the common ratio
between the values.
Plug in known values
Simplify
an = a1 * r n-1

10. Let’s try one
Let’s try one
Find the 10
Find the 10th
th
term in the sequence of
term in the sequence of
1, -6, 36, -216 . . .
1, -6, 36, -216 . . .
a10 = 1(-6)9
= -10,077,696
Common ratio = -6
a10 = 1 (-6) (10-1)
Start with the explicit sequence formula
Find the common ratio
between the values.
Plug in known values
Simplify
an = a1 * r n-1

11. Example
Example
 Find the 7
Find the 7th
th
term of a geometric
term of a geometric
progression with a 1
progression with a 1st
st
term of 3 and
term of 3 and
a common ratio of 1/2?
a common ratio of 1/2?
an = a1 * r n-1
a7 = 3 * 1/2 7-1
a7 = 3 * 1/2 6
a7 = 3 * 1/64
a7 = 3/64

12. Ex.
Ex.
Write the first five terms of the
geometric sequence whose first
term is a1 = 9 and r = (1/3).
9 3 1
1
3
1
9
, , , ,

13. Ex.
Ex.
Find the 15
Find the 15th
th
term of the geometric
term of the geometric
sequence whose first term is 20 and
sequence whose first term is 20 and
whose common ratio is 1.05
whose common ratio is 1.05
an = a1rn – 1
a15 = (20)(1.05)15 – 1
a15 = 39.599

14. Ex. 4
Ex. 4
Find a formula for the nth term.
What is the 9th term?
5, 15, 45, …
an = 5(3)n – 1
an = 5(3)n – 1
a9 = 5(3)8
a9 = 32805
an = a1rn – 1

15. Example
Example
 Find the 6
Find the 6th
th
term of a geometric
term of a geometric
progression with a 1
progression with a 1st
st
term of 2 and
term of 2 and
a common ratio of 1/3?
a common ratio of 1/3?
an = a1 * r n-1
a6 = 2 * 1/3 6-1
a6 = 2 * 1/3 5
a6 = 2 * 1/243
a6 = 2/243

16. Ex. 5
Ex. 5
The fourth term of a geometric sequence is
The fourth term of a geometric sequence is
125, and the 10
125, and the 10th
th
term is 125/64. Find the
term is 125/64. Find the
14
14th
th
term. (Assume all terms are positive)
term. (Assume all terms are positive)
a4 = 125
a10 =
125
64
125
64
125 6
 r
a10 = a4r6
1
64
6
r
1
2
r
a14 = a10r4
125
1024
a14 =
4
14
2
1
64
125







a