2, 4, 8, 16, … 32Exercise

2, 4, 6, 8, … Exercise

10

1, 3, 9, 27, … 81Exercise

1, , , , … 12

14

18

116

Exercise

1, –2, 4, –8, 16, … –32Exercise

3 6

x 2

12

x 2

24

x 2

48

x 2

96

x 2

Geometric SequenceA geometric sequence is a sequence of numbers whose successive terms differ by a constant multiplier.

Common RatioThe constant multiplier for a geometric sequence is called the common ratio, r.

State whether the sequence 8, 4, 2, 1, … is arithmetic or geometric.

geometric

Example

State whether the sequence –6, –18, –54, –162, … is arithmetic or geometric.

geometric

Example

State whether the sequence 5, 7, 9, 11, … is arithmetic or geometric.

arithmetic

Example

State whether the sequence 5, 10, 20, 40, … is arithmetic or geometric.

geometric

Example

Geometric SequenceTerms differ by a constant factor r.

an = an – 1r

Write the first six terms of the geometric sequence in which a1 = 1 and r = 3.a1 = 1a2 = 1 • 3 = 3a3 = 3 • 3 = 9a4 = 9 • 3 = 27

Example 1

The first six terms of the sequence are

1, 3, 9, 27, 81, and 243.

Write the first six terms of the geometric sequence in which a1 = 1 and r = 3.

Example 1

a5 = 27 • 3 = 81a6 = 81 • 3 = 243

Find the value of a1 for the sequence 2, 6, 18, 54, 162, 486, …

a1 = 2

Example 2

Find the value of r for the sequence 2, 6, 18, 54, 162, 486, …

r = 3

Example 2

Find the value of a3 for the sequence 2, 6, 18, 54, 162, 486, …

a3 = 18

Example 2

Find the value of a8 for the sequence 2, 6, 18, 54, 162, 486, …

= 4,374

a7 = 486 • 3= 1,458

a8 = 1,458 • 3

Example 2

Geometric SequenceTerms differ by a constant factor r.

an = an – 1r

Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1.a1 = –4a2 = 3(–4) = –12a3 = 3(–12) = –36a4 = 3(–36) = –108a5 = 3(–108) = –324

Example 3

Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1.

The first five terms of the sequence are

–4, –12, –36, –108, and –324.

Example 3

Write the first four terms of the sequence defined bya1 = 2 and an = 4an – 1.

2, 8, 32, 128

Example

Write the first four terms of the sequence defined bya1 = –3 and an = 2an – 1.

–3, –6, –12, –24

Example

Find the common ratio, r, of the sequence 4, –12, 36, –108.

r = –3

Example

Find the common ratio, r, of the sequence 24, 12, 6, 3.

r =

12

Example

Write the recursive formula for the sequence 729, 243, 81, 27, ...a1 = 729r =

13

an = an – 113

Example 4

3, 6 ,12, 24, 48, 96× 2 to get next term

1Position Term

23456n

33 • 21 = 63 • 22 = 123 • 23 = 243 • 24 = 483 • 25 = 963 • 2n – 1

Explicit FormulaThe explicit formula for a geometric sequence is an = a1r

n –1, in which a1 is the

first term and r is the common ratio.

Write the explicit formula for the sequence –5, –15, –45, –135, –405, ...a1 = –5r = 3 an = –5(3)n – 1

Example 5

Write the explicit formula for the sequence –3, –6, –12, –24, ...

an = –3(2)n – 1

Example

Write the explicit formula for the sequence 12, 6, 3, 1.5, ...

an = 12( )n – 112

Example

A ball bounces three-fourths the height of its fall. If the ball falls 12 ft., how high does it bounce on the first bounce? on the second bounce? on the third bounce?

9 ft.; 6.75 ft.; 5.0625 ft.

Exercise

In the last problem, the height of the bounces forms a geometric sequence. Find the common ratio of this geometric sequence.

r = 0.75

Exercise

If the ball falls 12 ft. and begins bouncing, what is the total distance it has traveled when it hits the ground the third time?

43.5 ft.

Exercise

When will the ball stop bouncing?

Exercise