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SEQUENCES AND SEQUENCES AND SERIES SERIES Arithmetic Arithmetic
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SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence. Finite Sequence: 2, 6, 10, 14 Finite Series:2.

Jan 02, 2016

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Page 1: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

SEQUENCES AND SEQUENCES AND SERIESSERIES

ArithmeticArithmetic

Page 2: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

DefinitionDefinitionA A seriesseries is an indicated sum of the terms of a is an indicated sum of the terms of a

sequence.sequence.

Finite Sequence: Finite Sequence: 2, 6, 10, 142, 6, 10, 14

Finite Series:Finite Series: 2 + 6 + 10 + 142 + 6 + 10 + 14

Page 3: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

FormulaFormula

The sum of the first The sum of the first nn terms of an terms of an

arithmetic series is: arithmetic series is:

1 n

n

n a aS

2

Page 4: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

ExampleExample1.1. Find SFind S2525 of the arithmetic series 11 + 14 + of the arithmetic series 11 + 14 +

17 + 20+ …17 + 20+ …

Page 5: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

ExampleExample1.1. Find SFind S2525 of the arithmetic series 11 + 14 + of the arithmetic series 11 + 14 +

17 + 20+ …17 + 20+ …

n 1

n

25

25

a a d n 1

a 11 3 n 1

a 11 3 25 1

a 83

First you have to find the 25th term.

Now you can find the sum of the first 25 terms.

1 nn

25

25

n a aS

225 11 83

S2

S 1175

Page 6: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Examples ContinuedExamples Continued

2.2. Find the sum of the arithmetic series Find the sum of the arithmetic series 5+9+13+…+153.5+9+13+…+153.

Page 7: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Examples ContinuedExamples Continued

2.2. Find the sum of the arithmetic series Find the sum of the arithmetic series 5+9+13+…+153.5+9+13+…+153.

n 1a a d n 1

153 5 4 n 1

148 4 n 1

37 n 1

38 n

First you must determine how many terms you are adding.

1 nn

38

38

n a aS

238 5 153

S2

S 3002

Now you can find the sum of the first 38 terms.

Page 8: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

SeriesSeriesThe sum of the first n terms of a geometric The sum of the first n terms of a geometric

series is:series is:

n1

n

a 1- rS =

1- r

Page 9: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

ExamplesExamples1.1. Find the sum of the first 10 terms of the Find the sum of the first 10 terms of the

geometric series 2 – 6 + 18 – 54 +…geometric series 2 – 6 + 18 – 54 +…

Page 10: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

ExamplesExamples1.1. Find the sum of the first 10 terms of the Find the sum of the first 10 terms of the

geometric series 2 – 6 + 18 – 54 +…geometric series 2 – 6 + 18 – 54 +…

n1

n

10

10

10

a 1 rS

1 r

2 1 2S

1 2

S 29,524

Page 11: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

INFINITE SERIESINFINITE SERIES

GeometricGeometric

Page 12: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Converge Vs. DivergeConverge Vs. Diverge An infinite series converges if the ratio An infinite series converges if the ratio

lies between -1 and 1.lies between -1 and 1. Do the following series converge or Do the following series converge or

diverge?diverge?

1.1. 3+9+27+…3+9+27+…

2.2. 16+4+1+1/4+…16+4+1+1/4+…

Page 13: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

FormulaFormula

The sum, S, of an infinite geometric series The sum, S, of an infinite geometric series

where -1<r<1 is given by the following where -1<r<1 is given by the following

formula:formula:

1aS =

1- r

Page 14: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

ExamplesExamplesFind SFind S11, S, S22, S, S33, S, S44, and the infinite sum if it , and the infinite sum if it

exists.exists.

1.1. 8+4+2+1+…8+4+2+1+… 2.2. 1+3+9+27+… 1+3+9+27+…

Page 15: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

ExamplesExamplesFind SFind S11, S, S22, S, S33, S, S44, and the infinite sum if it , and the infinite sum if it

exists.exists.

1.1. 8+4+2+1+…8+4+2+1+… 2.2. 1+3+9+27+… 1+3+9+27+…

1

2

3

4

1

S 8

S 8 4 12

S 8 4 2 14

S 8 4 2 1 15

a 8S 16

11 r 12

1

2

3

4

S 1

S 1 3 4

S 1 3 9 13

S 1 3 9 27 40

Series Diverges

Page 16: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

SIGMA NOTATIONSIGMA NOTATION

Page 17: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Sigma NotationSigma Notation

Consider the series:Consider the series:

6 + 12 + 18 + 24 + 30 6 + 12 + 18 + 24 + 30

or another way,or another way,

6(1) + 6(2) + 6(3) + 6(4) + 6(5) = 6n6(1) + 6(2) + 6(3) + 6(4) + 6(5) = 6n

Page 18: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Sigma NotationSigma Notation

5

n 1

6n

In Sigma Notation, that series would look like:In Sigma Notation, that series would look like:

It is read as “the sum of 6n for values of n from 1 to 5”

Page 19: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

PARTS OF SIGMAPARTS OF SIGMA

5

n 1

6n

Summand – 6n

Index – n

Lower Limit – 1

Upper Limit – 5

Page 20: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Sigma Notation Sigma Notation ContinuedContinued

The lower limit is the first number that is The lower limit is the first number that is

being substituted in for n.being substituted in for n.

The upper limit is the last number that is The upper limit is the last number that is

being substituted in for n.being substituted in for n.

Page 21: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

EXAMPLESEXAMPLESWrite in expanded form and find the Write in expanded form and find the

sum:sum:

1. 1. 2.2.

4

n 1

3n 7

4

n 1

3n 6

Page 22: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

EXAMPLESEXAMPLESWrite in expanded form and find the Write in expanded form and find the

sum:sum:

1. 1. 2.2.

4

n 1

3n 7

4

n 1

3n 6

3 1 3 2 3 3 3 4 7

3 6 9 12 7

37

Lower Limit Upper Limit

3 1 6 3 2 6 3 3 6 3 4 6

3 0 3 6

6

Upper LimitLower Limit

Page 23: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Express using Sigma Notation:Express using Sigma Notation:

3. 10 + 15 + 20 + … + 1003. 10 + 15 + 20 + … + 100

4. 5 + 10 +20 + 40 + 80 + 1604. 5 + 10 +20 + 40 + 80 + 160

Page 24: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Express using Sigma Notation:Express using Sigma Notation:

3. 10 + 15 + 20 + … + 1003. 10 + 15 + 20 + … + 100

4. 5 + 10 +20 + 40 + 80 + 1604. 5 + 10 +20 + 40 + 80 + 160

19

n 1

5 5n

Page 25: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

Express using Sigma Notation:Express using Sigma Notation:

3. 10 + 15 + 20 + … + 1003. 10 + 15 + 20 + … + 100

4. 5 + 10 +20 + 40 + 80 + 1604. 5 + 10 +20 + 40 + 80 + 160

19

n 1

5 5n

6

n 1

n 1

5 * 2

Page 26: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

5. 1 + 4 + 9 + 16 + 25 + …5. 1 + 4 + 9 + 16 + 25 + …

6. 1 + ½ + 1/3 + ¼ + …6. 1 + ½ + 1/3 + ¼ + …

Page 27: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

5. 1 + 4 + 9 + 16 + 25 + …5. 1 + 4 + 9 + 16 + 25 + …

6. 1 + ½ + 1/3 + ¼ + …6. 1 + ½ + 1/3 + ¼ + …

2

n 1

n

Page 28: SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

5. 1 + 4 + 9 + 16 + 25 + …5. 1 + 4 + 9 + 16 + 25 + …

6. 1 + ½ + 1/3 + ¼ + …6. 1 + ½ + 1/3 + ¼ + …

2

n 1

n

n 1

1n