2, 4, 8, 16, … 32 Exercise

2, 4, 6, 8, … Exercise 10

1, 3, 9, 27, … 81 Exercise

1,,,, … Exercise

1, –2, 4, –8, 16, … –32 Exercise

x 2 12 x 2 24 x 2 48 x 2 96 x 2

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

Common Ratio The 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 Sequence Terms differ by a constant factor r. a n = a n – 1 r

Write the first six terms of the geometric sequence in which a 1 = 1 and r = 3. a 1 = 1 a 2 = 1 3 = 3 a 3 = 3 3 = 9 a 4 = 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 a 1 = 1 and r = 3. Example 1 a 5 = 27 3 = 81 a 6 = 81 3 = 243

Find the value of a 1 for the sequence 2, 6, 18, 54, 162, 486, … a 1 = 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 a 3 for the sequence 2, 6, 18, 54, 162, 486, … a 3 = 18 Example 2

Find the value of a 8 for the sequence 2, 6, 18, 54, 162, 486, … = 4,374 a 7 = = 1,458 a 8 = 1,458 3 Example 2

Geometric Sequence Terms differ by a constant factor r. a n = a n – 1 r

Write the first five terms of the sequence defined by a 1 = –4 and a n = 3a n – 1. a 1 = –4 a 2 = 3(–4) = –12 a 3 = 3(–12) = –36 a 4 = 3(–36) = –108 a 5 = 3(–108) = –324 Example 3

Write the first five terms of the sequence defined by a 1 = –4 and a n = 3a n – 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 by a 1 = 2 and a n = 4a n – 1. 2, 8, 32, 128 Example

Write the first four terms of the sequence defined by a 1 = –3 and a n = 2a n – 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 = Example

Write the recursive formula for the sequence 729, 243, 81, 27,... a 1 = 729 r = a n = a n – Example 4

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

1 Position Term n = = = = = n – 1

Explicit Formula The explicit formula for a geometric sequence is a n = a 1 r n –1, in which a 1 is the first term and r is the common ratio.

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

Write the explicit formula for the sequence –3, –6, –12, –24,... a n = –3(2) n – 1 Example

Write the explicit formula for the sequence 12, 6, 3, 1.5,... a n = 12( ) n – 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.; 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