What you really need to know! A geometric sequence is a sequence in which the quotient of any two...

What you really need to know!

A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 256, . . , the common ratio is 4.

GeometricGeometric SeriesSeries

Geometric SequenceGeometric Sequence

• The ratio of a term to it’s The ratio of a term to it’s previous term is constant.previous term is constant.

• This means you multiply by the This means you multiply by the same number to get each term.same number to get each term.

• This number that you multiply by This number that you multiply by is called the is called the common ratiocommon ratio ( (rr).).

ExampleExample: Decide whether each : Decide whether each sequence is geometric.sequence is geometric.

• 4,-8,16,-32,…

• -8/4=-2

• 16/-8=-2

• -32/16=-2

• Geometric (common ratio is -2)

• 3,9,-27,-81,243,…

• 9/3=3

• -27/9=-3

• -81/-27=3

• 243/-81=-3

• Not geometric

Rule for a Geometric Rule for a Geometric SequenceSequenceuunn=u=u11r r n-1n-1

ExampleExample: Write a rule for the n: Write a rule for the nth th

term of the sequence 5, 2, 0.8, 0.32,term of the sequence 5, 2, 0.8, 0.32,… . Then find u… . Then find u88..

•First, find r.First, find r.

•r= r= 22//5 5 = 0.4= 0.4

•uunn=5(0.4)=5(0.4)n-1n-1

uu88=5(0.4)=5(0.4)8-18-1

uu88=5(0.4)=5(0.4)77

uu88=5(0.0016384)=5(0.0016384)

uu88=0.008192=0.008192

One term of a geometric sequence is One term of a geometric sequence is uu4 4 = 3= 3. .

The common ratio is The common ratio is r = 3r = 3. Write a rule for the . Write a rule for the nth nth term. Then graph the sequence.term. Then graph the sequence.

• If uIf u44=3, then when =3, then when n=4, un=4, unn=3.=3.

• Use uUse unn=u=u11rrn-1n-1

3=u3=u11(3)(3)4-14-1

3=u3=u11(3)(3)33

3=u3=u11(27)(27)11//99=a=a11

• uunn=u=u11rrn-1n-1

uunn=(=(11//99)(3))(3)n-1n-1

• To graph, graph To graph, graph the points of the points of the form (n,uthe form (n,unn).).

• Such as, (1,Such as, (1,11//99), ), (2,(2,11//33), (3,1), ), (3,1), (4,3),…(4,3),…

Please find the 15Please find the 15thth term term

• 5, 10, 20, 40

• So, geometric sequence with u1 = 5 r = 2

10 20 402

5 10 20

11n

nu u r 15 2nnu x

1415 5 2 81,920u x

Compound Interest

8 - 9

100110

121

1000

1210

1331

1100100 100

110

Time(Years)0 1 2 3 4

Amount $1000

Amount $1000

110

InterestInterestInterestInterest

100

InterestInterest

133.1

Compounding Period

Compounding Period

Compounding Period

Compounding Period

InterestInterest

121

Compound Interest- Future Value

COMPOUND INTEREST FORMULA

FVFV is the is the Future Value Future Value in in tt yearsyears

PP is the is the Present Value Present Value amount started with amount started with r r is the annual interest rate is the annual interest rate

nn number of times it compounds per year. number of times it compounds per year.

1nt

rFV P

n

Find the amount that results from the Find the amount that results from the investment:investment:

$50 invested at 6% compounded monthly $50 invested at 6% compounded monthly after a period of 3 years.after a period of 3 years.

EXAMPLEEXAMPLE

)3(12

1 2.0 6

1 5 0 F V $59.83$59.83

Investing $1,000 at a rate of 10% Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, compounded annually, quarterly, monthly, and daily will yield the following amounts and daily will yield the following amounts after 1 year:after 1 year:

FV = PV(1 + r) = 1,000(1 + .1) = $1100.00FV = PV(1 + r) = 1,000(1 + .1) = $1100.00

COMPARING COMPARING COMPOUNDING PERIODSCOMPOUNDING PERIODS

$ 1 1 0 3 .8 1 4.1

1 1 0 0 0 F V

4

Investing $1,000 at a rate of 10% Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, compounded annually, quarterly, monthly, and daily will yield the following amounts and daily will yield the following amounts after 1 year:after 1 year:

COMPARING COMPARING COMPOUNDING PERIODSCOMPOUNDING PERIODS

$ 1 1 0 4 .7 1 1 2.1

1 1 0 0 0 F V

12

$ 1 1 0 5 .1 6 3 6 5

.1 1 1 0 0 0 F V

365

Investing $1,000 at a rate of 10% Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, compounded annually, quarterly, monthly, and daily will yield the following amounts and daily will yield the following amounts after 1 year:after 1 year:

Interest EarnedInterest Earned

$ 1 0 4 . 7 1 $ 1 0 0 0- $ 1 1 0 4 . 7 1 E a r n e d I n t

$ 1 0 5 .1 6 $ 1 0 0 0- $ 1 1 0 5 .1 6 I .E .

HomeworkHomework

Sum of a Finite Geometric Series

• The sum of the first n terms of a geometric series is

1(1 )

1

n

n

a rS

r

Notice – no last term needed!!!!

Formula for the Sum of a Finite Formula for the Sum of a Finite Geometric SeriesGeometric Series

r

raS

n

n 1

11

n = # of termsn = # of terms

aa1 1 = 1= 1stst term term

r = common ratior = common ratio

What is n? What is a1? What is r?

Example

• Find the sum of the 1st 10 terms of the geometric sequence: 2 ,-6, 18, -54

10 10

10

2(1 - (-3) ) 2(1 - 3 )S = =

1- -3 29,524

4

1(1 )

1

n

n

a rS

rWhat is n? What is a1? What is r?

That’s It!

Example: Consider the geometric Example: Consider the geometric series 4+2+1+½+… .series 4+2+1+½+… .

• Find the sum of the first 10 terms.

r

raS

n

n 1

11

21

1

21

14

10

10S

128

1023

1024

20464

21

10241023

4

211024

11

410

S

GeometricGeometric SeriesSeries

Infinite Geometric Series

• Consider the infinite geometric sequence

•

• What happens to each term in the series?

• They get smaller and smaller, but how small does a term actually get?

1 1 1 1 1, , , ,... ...

2 4 8 16 2

n

Each term approaches 0

Infinite Sum

1

1 What is the area of the square?

Cut the square in half and label the area of one section.

Cut the unlabeled area in half and label the area of one

section.

Continue the process…

Sum all of the areas:

1 1 1 1 1 1 12 4 8 16 32 64 128 ... 1

2...n

The general term is…1

Since the infinite sum represents the area of the square…

12

1n

n

Infinite Series

Connecting Series and Sequences

Find the sum of…

Partial Sums of a Series

Convergent or Divergent Series

Convergent or Divergent Series

Examples

Why do 1, 3 Diverge?

Arithmetic and Geometric Series

Definition of a Geometric Series

Partial Sums

• Look at the sequence of partial sums:

1

2

3

121 1 32 4 41 1 1 72 4 8 8

S

S

S

What is happening to the

sum?

It is approaching 1

0

1

It’s CONVERGING

TO 1.

Here’s the Rule

Sum of an Infinite Geometric Sum of an Infinite Geometric SeriesSeries

If If |r| < 1|r| < 1, the infinite geometric series, the infinite geometric series

aa11 + a + a11r + ar + a11rr22 + … + a + … + a11rrn n + …+ …

converges to the sumconverges to the sum

If If |r| > 1|r| > 1, then the series diverges , then the series diverges (does not have a sum)(does not have a sum)

1

1

aS

r

Converging – Has a Sum

• So, if -1 < r < 1, then the series will converge. Look at the series given by

• Since r = , we know that the sum

is

• The graph confirms:

1 1 1 1

...4 16 64 256

1

4

1

114

11 31

4

aS

r

Diverging – Has NO Sum

• If r > 1, the series will diverge. Look at 1 + 2 + 4 + 8 + ….

• Since r = 2, we know that the series grows without bound and has no sum.

• The graph confirms:

1

2

3

1

1 2 3

1 2 4 7...

S

S

S

Example

• Find the sum of the infinite geometric series 9 – 6 + 4 - …

• We know: a1 = 9 and r = ?

2

3

1 9 2721 51

3

aS

r

You Try

• Find the sum of the infinite geometric series 24 – 12 + 6 – 3 + …

• Since r = -½ 1

124 24 48

161 3 31

2 2

aS

r

S

1

3 3 31. 0.3 ....

10 100 1000

.) Find i. ii. a u r

1

3.) i.

10a u

31100

.) ii. r 0.103 10

10

a

3 3 31. 0.3 ....

10 100 1000

1.) Using , show that 0.a 3

3b

3 3 3.) 0.3 ...

10 100 1000

3 3 30.3 ...

10 100 1000n

b

S

1

310, then

10

1 1

1u

n Sr

10.3

3