Download 11.3 – Geometric Sequences

11.3 – Geometric
Sequences
What is a Geometric Sequence?
 In
a geometric sequence, the ratio
between consecutive terms is
constant. This ratio is called the
common ratio.
 Unlike in an arithmetic sequence, the
difference between consecutive
terms varies.
 We look for multiplication to
identify geometric sequences.
Ex: Determine if the sequence is geometric.
If so, identify the common ratio
 1,
-6, 36, -216
yes. Common ratio=-6
 2,
4, 6, 8
no. No common ratio
Important Formulas for
Geometric Sequence:

Recursive Formula
an = (an – 1 ) r
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

Explicit Formula
an = a1 * r n-1

Geometric Mean
Find the product of
the two values and
then take the
square root of the
answer.
Let’s start with the geometric mean
 Find
the geometric mean between
3 and 48
Let’s try one: Find the geometric mean
between 28 and 5103
Ex: Write the explicit formula for
each sequence
First term: a1 = 7
Common ratio = 1/3
Explicit:
an = a1 * r n-1
Now find the first five
terms:
a1
a2
a3
a4
a5
=
=
=
=
=
7(1/3)
7(1/3)
7(1/3)
7(1/3)
7(1/3)
(1-1)
(2-1)
(3-1)
(4-1)
(5-1)
=
=
=
=
=
7
7/3
7/9
7/27
7/81
Explicit Arithmetic Sequence Problem
Find the 19th term in the sequence of
11,33,99,297 . . .
an = a1 * r n-1
Common ratio = 3
Start with the explicit sequence formula
Find the common ratio
between the values.
a19 = 11 (3) (19-1)
a19 = 11(3)18 =4,261,626,379
Plug in known values
Simplify
Student Check
Find the 10th term in the sequence of
1, -6, 36, -216 . . .
an = a1 * r n-1
Start with the explicit sequence formula
Common ratio = -6
a10 = 1 (-6) (10-1)
a10 = 1(-6)9 = -10,077,696
Find the common ratio
between the values.
Plug in known values
Simplify
Word Problems:
 If
the number of bacteria in a colony
doubles every five days and on Jan.
16th there were 8,000,000 bacteria,
how many bacteria were there on
Jan 6th?
Word Problem
A
women made $35,000 on the first
year of her new job. Each year she
receives a 10% raise. How much did
she earn on the tenth year of her
job?