Geometric Sequences

A sequence which has a constant ratio between terms. The rule is exponential.

Example: 4, 8, 16, 32, 64, …

(generator is x2)

Geometric Sequences

n t(n)

1 4

2 8

3 16

4 32

5 64

x2

x2

x2

x2 2 2n

t n

Discrete

0 1 2 3 4 5 6

First find the generator and the n=0 term. Then write the equation:

Ex:

Working Backwards for a Rule

1 2 3 43,

5

0

35

5n

t n

3, 15, 75, 375, …x5

Seq

uenc

es s

tart

with

n=

1 no

w!

t(0) is not in the sequence! Do not include it in tables or graphs!

The sequence increases.

Positive Multipliers

In a geometric sequence, if the multiplier is:• Less than one but greater than 0 (0<b<1)

• Equal to 1 (b=1)

• Greater than 1 (b>1)

The sequence decreases.

The sequence is constant.

Example of a Sequence

2, 6, 18, 54, ___, ___, …Generator:

Representations

Table: Rule:

Multiply by 3

162 486

n t(n)

1 2

2 6

3 18

4 54

5 162

6 486

t(n) = 2/3(3)n

n t(n)

1

2

3

4

5

6

Example of a Sequence

625, 125, 25, 5, ___, ___, …

Generator:

Representations

Table: Rule:

Multiply by 1/5 (0.2)

1 0.2

n t(n)

1 625

2 125

3 25

4 5

5 1

6 0.2

t(n) = 3125(0.2)n

n t(n)

1

2

3

4

5

6

Sequence: t(n)

Function: f(x)

Sequences v Functions

Positive Integers (sometimes 0)

Can be all Real numbers

Domain (n) =

Domain (x) =

Can be all Real numbers

Range (f(x))=

DiscreteThe Graph is

can be ContinuousThe Graph

Range (t(n)) =

Can be all Real numbers