SEQUENCES AND SEQUENCES AND SERIESSERIES

ArithmeticArithmetic

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

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

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

17 + 20+ …17 + 20+ …

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

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.

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.

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

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 +…

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

INFINITE SERIESINFINITE SERIES

GeometricGeometric

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+…

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

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+…

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

SIGMA NOTATIONSIGMA NOTATION

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

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”

PARTS OF SIGMAPARTS OF SIGMA

5

n 1

6n

Summand – 6n

Index – n

Lower Limit – 1

Upper Limit – 5

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.

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

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

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

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

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

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

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

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

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