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SIGMA NOTATIOM, SEQUANCES AND SERIES

MAIN TOPIC

4.1 Sequences and Series4.2 Sigma Notation4.3 Arithmetic Sequences and Series4.3 Geometric Sequences and Series4.4 Infinite Geometric Series

OBJECTIVE

At the end of this topic you should be able to Define sequences and series Understand finite and infinite sequence,

finite and infinite series Use the sum notation to write a series

4

SEQUENCES and SERIES

A sequence is a set of real numbers a1, a2,…an,… which is arranged (ordered).

Example:

Each number ak is a term of the sequence.

We called a1 - First term and a45 - Forty-fifth term

The nth term an is called the general term of the sequence.

3,9,27, ,3 ,n

INFINITE SEQUENCES

An infinite sequence is often defined by stating a formula for the nth term, an by using {an}.

Example:

The sequence has nth term . Using the sequence notation, we write this

sequence as follows 1 2 3 152 ,2 ,2 , , 2 , 2 ,n

{2 }n 2nna

First three terms Fifth teen term

INFINITE SEQUENCES

6

EXERCISE 1 : Finding terms of a sequence

List the first four terms and tenth term of each sequence:

3n n

2

11

3 1n n

n

1

n

n

2 0.1n

A

B

C

D

E

F 4 2 1n

Definition of Sigma Notation

Consider the following addition :

Based on the patterns of the addends, the addition above can be written in the following form

2 + 5 + 8 + 11 + 14

=(3(1) – 1) + (3(2) – 1) + (3(3) – 1) + (3(4) – 1) + (3(5) – 1) + (3(6) – 1)

2 + 5 + 8 + 11 + 14

The amount of the term in the addition above can be written as (3i – 1). The term in the addition are obtained by substituting the value of i with the value of 1, 2, 3, 4, and 5 to (3i – 1)

The symbol read as sigma, is used to simplify the expression of the addition of number with certain patterns. In order that you understand more, the addition above can be written as := 2 + 5 + 8 + 11 + 14 =(3(1) – 1) + (3(2) – 1) + (3(3) – 1) + (3(4) – 1) + (3(5) – 1) + (3(6)-1)

=(3(1) – 1) + (3(2) – 1) + … + (3(i) – 1) + … + (3(6)-1)

6

1i1) (3i

9

In general, the sum of n term of number with certain pattern where the ith term is stated as Ui can be written as :

U1 + U2 + U3 + … + Ui + …+ Un =

n

1ii

U

Where :i = 1 is the lower bound of the additionn is the upper bound of the addition

Example 1 :

Change the following of addition in the form of sigma notation! a. 3 + 6 + 9 + 12 + 15 + 18 + 21

b. 1 + 5 + 9 + 13 + 17 + 21

Answer :

a. 3 + 6 + 9 + 12 + 15 + 18 + 21

= 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6) + 3(7)

= 3(1) + 3(2) + … + 3(i) + … + 3(7)

=

b. 1 + 5 + 9 + 13 + 17 + 21

= (4(1) – 3) + (4(2) – 3) + (4(3) – 3) + … + (4(6) – 3)

= (4(i) – 3)

=

7

1i(3i)

6

1i3i4

Determine the value of the following addition that state in sigma notation!

a.

b.

4

1i

3) (2i

6

1i

1) (i2

Answer :

4

1i

3) (2i

6

1i

1) (i2

161514131211 222222

THEOREM OF SUMS

1 1 1

n n n

k k k kk k k

a b a b

1 1 1

n n n

k k k kk k k

a b a b

1 1

n n

k kk k

ca c a

1

n

k

c nc

1n

k m

c n m c

Sum of a constant

Sum of 2 infinite sequences

Arithmetic Sequence and Series

OBJECTIVE

At the end of this topic you should be able to :Recognize arithmetic sequences and seriesDetermine the nth term of an arithmetic

sequences and series Recognize and prove arithmetic mean of an

arithmetic sequence of three consecutive terms a, b and c

THE nth TERM OF AN ARITHMETIC SEQUENCES

An arithmetic sequence with first term a and common different b, can be written as follows:

The nth term, an of this sequence is given by the following formula:

a, a + b, a + 2b, … , a + (n – 1)b

Un = a + (n – 1)b

ARITHMETIC SEQUENCES

A sequence U1, U2,…Un,… is an arithmetic sequence if there is a real number b such that for every positive integer k,

The number is called the common difference of the sequence.

U2 – U1 = U3 – U2 = … = Un – Un-1 = a constant

b = U2 – U1

Example 1:

Find the formula for the and term if given the arithmetic sequence :

A. 1, 4, 7, 10, …

B. 53, 48, 43, …

thn

Answer :

A 1, 4, 7, 10 Based on the sequence, then obtained :a = 1, b = U2 – U1 = 4 – 1 = 3Un = a + (n – 1)b = 1 + (n – 1)3 = 1 + 3n – 1 = 3nU30 = 3n = 3(30) = 90

Answer :

B 53, 48, 43, …Based on the sequence, then obtained :a = 53, b = U2 – U1 = 48 – 53 = -5Un = a + (n – 1)b = 48 + (n – 1)-5 = 48 + (-5n + 5) = 48 – 5n + 8 = 53 – 5nU30 = 53 – 5n = 53 – 5(30) = 53 – 150 = -97

EXERCISE 9: Finding a specific term of an arithmetic sequence

A. If given the arithmetic sequence U6 = 50 and U41 = 155 determine the twelfth.

B. If the fourth term of an arithmetic sequence is 5 and the ninth term is 20, find the sixth term.

Answer :

A U6 = 50, U21 = 155, U12? U6 = a + 5b = 50 U21 = a + 20b = 155

-15b = -105 b = 7

a + 5b = 50 a + 5(7) = 50 a = 50 - 35 a = 15U30 = a + (n – 1)b = 15 + (29)7 = 15 + 203 = 218

Answer :

A U4 = 5, U9 = 20, U6? U4 = a + 3b = 5 U9 = a + 8b = 20

-5b = -15 b = 3

a + 3b = 5 a + 3(3) = 5 a = 5 - 9 a = -4U6 = a + (n – 1)b = -4 + (5)3 = -4 + 15 = 11

Formula for the middle Term of on Arithmetic Sequence

2UUU n1

t

Example :

Given an arithmetic sequence of 3, 8, 13, … , 283. Determine the middle term of the sequence. Which term is the middle term

Answer :

From the question it is know that U1 = 3 and Un = 283

So the middle term is Ut = 143 Ut = 143

a + (t – 1)b = 143 3 + (t – 1)5 = 143

(t – 1)5 = 140t – 1 = 28 t = 29

1432

28622833

2UUU n 1

t

THE nth PARTIAL SUM OF AN ARITHMETIC SEQUENCES

If a1, a2,…an,… is an arithmetic sequence with common difference b, then the nth partial sum Sn (that is the sum of the first nth terms) is given by either

or

bnan

Sn 122

Unan

Sn 2

1 nnn SSU

OBJECTIVE

At the end of this topic you should be able to

Recognize geometric sequences and series Determine the nth term of a geometric sequences and series Recognize and prove geometric mean of an geometric sequence of

three consecutive terms a, b and c Derive and apply the summation formula for infinite geometric series Determine the simplest fractional form of a repeated decimal number

written as infinite geometric series

GEOMETRIC SEQUENCES

A sequence U1, U2,…Un,… is a geometric sequence if U1 ≠ 0 .

The number is called the common ratio of

the sequence.

1

n

n

U

Ur

1n32n4321 ar,...,ar,arar,a,or U,...,U,U,U,U

1nn arU

Example :

Given Geometric sequence of 24, 12, 6, 3, …, where the formula of the nth term is Un . Determine Un and the sixth term of the sequence

Answer:

The geometric sequence of 24, 12, 6, 3, … , the first term of the sequence a = 24 and the ratio r= ½ . The formula for the nth term is :

n

n

n

4

13

1

1nn

2.3

2.3.2

2

1.24

arU

4

3

3.2

2.3

3.2U

2-

64

n4n

THE SUM OF AN INFINITE GEOMETRIC SERIES

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

has the sum 1

1n

aS

r

2 3 11 1 1 1 1, ,na a r a r a r a r

THE nth PARTIAL SUM OF AN GEOMETRIC SEQUENCES

r1

)r(1 as

n

n

1r

1)-(r as

n

n

Example :

Given that a geometric sequence : 2, 6, 18, 54, …, Un, Determine :a. The formula for nth term andb. The sum of the sixth n term!

Answer

The formula for nth term is

Sequence a geometric sequence : 2, 6, 18, 54, … UnHave a = 2 and r = 3

1-n2.3

arU 1nn

The Sum of the first n term is :

26 2

2(26)

13

)13(2

1

1S

3

n

r

ra n

THE SUM OF AN INFINITE GEOMETRIC SERIES

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

has the sum1

1n

aS

r

2 3 11 1 1 1 1, ,na a r a r a r a r

EXERCISE 17: Find the sum of infinite geometric series

A. The following sequence is infinite geometric series. Find the sum

1

1

1 1 1 2) 2 ) 3

2 8 32 3

n

n

i ii

Answer :

i) a = 2 ,

....32

1

8

1

2

12

11

2 x

2

1

221

r

r

aS

1

12

2

)1(1

2

S

APPLICATIONS OF ARITHMETIC AND GEOMETRIC SERIES

OBJECTIVE

At the end of this topic you should be able to Solve problem involving arithmetic series Solve problem involving geometric series

42

APPLICATION 1: ARITHMETIC SEQUENCE

A carpenter whishes to construct a ladder with nine rungs whose length decrease uniformly from 24 inches at the base to 18 inches at the top. Determine the lengths of the seven intermediate rungs.

a1 = 18 inches

a9 = 24 inches Figure 1

43

APPLICATION 2: ARITHMETIC SEQUENCE

The first ten rows of seating in a certain section of stadium have 30 seats, 32 seats, 34 seats, and so on. The eleventh through the twentieth rows contain 50 seats. Find the total number of seats in the section.

Figure 2

44

APPLICATION 1: GEOMETRIC SEQUENCE

A rubber ball drop from a height of 10 meters. Suppose it rebounds one-half the distance after each fall, as illustrated by the arrow in Figure 3. Find the total distance the ball travels.

5 5

10

1.25 1.25

2.5 2.5

Figure 3

45

APPLICATION 2: GEOMETRIC SEQUENCE

If deposits of RM100 is made on the first day of each month into an account that pays 6% interest per year compounded monthly, determine the amount in the account after 18 years. Figure 4