Chapter 8 Sec 3

Geometric Sequences and Series

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Pre Calculus Ch 8.3

Essential Question

How do you find terms and sums of geometric sequences?

Key Vocabulary:Geometric Sequence

Common ratio

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Pre Calculus Ch 8.3

Definition of a Geometric Sequence

Definition of a Geometric Sequence

A sequence is geometric if the ratio of consecutive terms are the same. So, the sequenceis geometric if there is a number r such that

The number r is the common ratio of the sequence.

,...,...,,, 4321 naaaaa

.0 ,...3

4

2

3

1

2 rra

a

a

a

a

a

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Pre Calculus Ch 8.3

Begin with n = 1. The common ratio is 2.

Example 1 a. The sequence whose nth term is 2n is geometric.

,...2 ... ,16 ,8 ,4 ,2 n

Begin with n = 1. The common ratio is 3.

b. The sequence whose nth term is 4(3n) is geometric..

,..., ... , , , n343241083612

c. The sequence whose nth term is n2 is NOT geometric.

2 2

4

same. not the are4

94

1

4

2

3

1

2 a

a and

a

a

 

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Pre Calculus Ch 8.3

The nth term of a Geometric Sequence

The nth term of a Geometric Sequence

The nth term of an geometric sequence has the form

Where r is the common ratio between consecutive terms of the sequence and every geometric sequence can be written in the following form.

If you know the nth term, you can find the (n + 1)th term by multiplying by r.

11

nn raa

,..., ..., a, a, a, a, aa n54321

,r, ..., ar,ar,arr, a, aa n 11

41

31

2111

raa nn 1

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Pre Calculus Ch 8.3

Example 2 Write the first five terms of the geometric sequence whose common ratio is r = 2 and whose first term is a1 = 3.

a1 = 3

a2 = 3(21)=6

a3 = 3(22)=12

a4 = 3(23)=24

a5 = 3(24)=48

1a

raa 12 2

13 raa 3

14 raa 4

15 raa

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Pre Calculus Ch 8.3

Example 3 The 15th term of the geometric sequence where a1 = 20 and r = 1.05.

an = a1 • r n – 1

a15 = 20 • (1.05)15 – 1

a15 = 20 • (1.05)14

60.3915 a

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Pre Calculus Ch 8.3

Example 4 Find a formula for the nth term of the following geometric sequence and what is the ninth term of the sequence?

5, 15, 45, … 35

15

1

2 a

ar

111 35 nn

n raa

199 35 a 835

805,3265615

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Pre Calculus Ch 8.3

Example 5 The 4th term of a geometric sequence is 125 and the 10th term is 125/64 . Find the 14th term. (Assume that the terms are all positive.)

9110 raa

61 rrrra

6

3

4

2

3

1

21 r

a

a

a

a

a

aa

6410 raa

41014 raa

612564

125r

6

64

1r

62

1

2

1r

4

14 2

1

64

125

a

16

1

64

125

1024

12514 a

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Pre Calculus Ch 8.3 The Sum of a Finite Geometric Sequence

The Sum of a Finite Geometric Sequence with common ratio r ≠ 1 is given by

Find the sum

.1

11

1

11

r

raraS

nn

i

in

.3.0412

1

n

n

1232112

1

3.04...3.043.043.043.04 n

n

a1 = 4(0.3), r = 0.3, and n = 12…

r

ra

n

n

n

1

13.04 1

12

1

71.1

3.01

3.013.04

12

When using this formula, be careful to check index begins at

i = 1. If not then you’ll need to adjust sum.

71.571.13.043.04 012

0

n

n 51.3.0471.13.04 112

2

n

n

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Pre Calculus Ch 8.3

Geometric Series

The Sum of a Infinite Geometric Series

If | r | < 1, then the infinite geometric series has the sum

Note: if | r | ≥ 1 the series has no sum.

The sum of an infinite geometric sequence is called infinite geometric series or geometric series.

If r has the property that | r | < 1, then it can be shown rn become arbitrarily close to zero as n increases without bound.

.1

1

01 r

araS

i

i

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Pre Calculus Ch 8.3

Example 7 Find the sixth partial sum of the series, then find the sum of the series.

a1 = 4 and r = 0.6

1

16.04n

n

1615141312116

1

1 6.046.046.046.046.046.046.04

n

n

53344.9311.518.864.44.14.24

106.01

46.04

1

1

n

n

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Pre Calculus Ch 8.3

Example 8 Find the sum 3 + 0.3 + 0.03 + 0.003 + …

a1 = 3 and 1.03

3.0

1

2 a

ar

r

aS

11 33.3

9.0

3

1.01

3

rtnt

PeAn

rPA

and 1

Compounded Interest Formulas

CompoundedDaily, Quarterly,…

CompoundedContinuously (only)

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Pre Calculus Ch 8.3

Essential Question

How do you find terms and

sums of geometric

sequences?

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Pre Calculus Ch 8.3

Daily Assignment

• Chapter 8 Section 3• Text Book

• Pg 607 – 608• All ODD OF:

• #1 – 21; #33 – 35; #55 – 65; #73 – 75; #83 – 85

• Read Section 8.5• Show all work for credit.