sequences and series

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SEQUENCE AND

SERIES ZONE

TO

ENTER

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For Senior High School

Standard Competence:

Using sequences and series concept to

solve problems

Motivation

Sequences Series

Refferences

Exit

Competence

(3)

references

Urban, Paul. 2004. Mathematics for International

Student.

Australia : Haese & Harris

Publications

Olive, Jenny. 2003. Maths a Student’s Survival

Guide.

United Kingdom : Cambridge

University Press

Motivation

Sequences Series

Refferences

Exit

Competence

Test

(4)

1. Determine nth terms sequences and number of n terms in arithmetic and geometry

2. Design mathematics model of problems related to sequences and series

3. Solve the mathematics model of problems related to sequences and series

Basic Competence

1. Explaining definition of sequences and series

2. Finding formula of arithmetic sequences and series 3. Finding formula of geometry sequences and series 4. Calculating nth terms and the number of n terms in

arithmetic and geometry series

5. Identifying problems related to series

6. Formulating mathematics model from series problems 7. Solving problems related to series

Indicators

(5)

NUMBER SEQUENCES

Motivation

Sequences

Series

Refferences

Exit

Competence

Test

(6)

Look at the picture below!!

MOTIVATION

For Senior High School

1 ,

Next

Back

Motivation

Sequences Series

Refferences

Exit

Competence

(7)

Consider to the illustrated of

amoeba. Every 5 minutes, amoeba

split self become 2 such that after

15 minutes they form this number

1, 2, 4, ...

Can you help me to determine how many

amoeba after 60minutes??????

That’s why we will learn this material.

Can you help me to determine how many

amoeba after 60minutes??????

That’s why we will learn this material.

For Senior High School

(8)

Let see here!!!

If you represents u_n as the number of bricks in row n (from the top) then

u₁ = 3, u₂ = 4, u₃ = 5, u₄ = 6,……… So,

The number pattern: 3,4,5,6…… is called a sequence of numbersYou can specified this sequences by using an explicit

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Definition:

A number sequences is a set of numbers defined by a rule for

positive integers.

Example:

3,5, 9,

7, 11,…..

U

_n

= {2n+1}

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(10)

Definition:

An arithmetic sequence is a sequence in which each term

differs from the previous one by the same fixed number

2, 5, 8,11,14,…..

Example

Is arithmetic sequence as 5 – 2 = 8 – 5 = 11 – 8 = 14 – 11

etc

For all positive

integer n where d

is a contant

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(11)

GENERAL FORMULA

Suppose the first term of an arithmetics

sequences is U

₁

and the common diference is

d.

Then,

.

It is the general term

for arithmetic

sequence with first

term U

₁

and common

diference

d

It is the general term

for arithmetic

sequence with first

term U

₁

and common

diference

d

For Senior High School

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(12)

Example:

1. Consider the sequence 2, 9, 16, 23,

30, ...

a. Show that the sequence is

arithmetic.

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b.

So, the general formula is

c. If n = 100 so

Solution:

a. 9 – 2 = 7

16 – 9 = 7

23 – 16 = 7

30 – 23 = 7

From the pattern we can find that the

common difference

d

is 7.

So, the sequence is arithmetics with U

₁

= 2

d

= 7

For Senior High School

(14)

Definition

Geometric sequence is a sequence that each term can be obtained from the previous one by multiplying by the same non zero constan.

Example

2, 4, 8, 16,32,…..

Is a geometric sequence as 2 x

2 = 4 , 4 x

2 = 8, 8

x

2 = 16 etc

For all positive

integer n where r is a ratio

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(15)

GENERAL FORMULA

Suppose the first term of an arithmetics

sequences is U

₁

and the common diference

is

d.

Then,

It is the general term

for arithmetic

sequence with first

term U

₁

and common

diference

d

.

For Senior High School

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(16)

For the sequence

a. Show that the sequence is geometric

b. Find the general term u

_n

c. Hence, find the 12

th

term as a fraction

Example

:

Solution:

The common ratio of the sequence is

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b.

Or

So,the general formula is

Or

c. If n = 12 so,

For Senior High School

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For Senior High School

ARITHMETICS SERIES

Motivation

Sequences

Series

Refferences

Author Exit

Competence

(19)

Definition:

An arithmetic series is the addition of successive terms of

an arithmetic sequence.

2, 5, 8, 11,….. , 14

Example:

Is an arithmetic

sequence.

So,

2 + 5 + 8 + 11 + ….. + 14

Is an

arithmetic

series.

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(20)

SUM OF AN ARITHMETIC SERIES

If the first term is u

₁

and the

common difference is

d,

then

the term are??????

u

_{1, ,}

u

₁

+ d , u

₁

+ 2d , u

₁

+ 3d ,

etc.

Now,

Suppose that u

_n

is the last term of an arithmetic

series.

Then,

Recall

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(21)

S

_n

= u

₁

+ (u

₁

+

d

) + (u

₁

+ 2

d

) + .... + (u

_n

- 2

d

) + (u

_n

–

_S

d

) + u

_n

n

= u

n

+ (u

n

-

d

) + (u

n

- 2

d

) + .... + (u

1

+ 2

d

)+ (u

1

+

d

) + u

₁

+

2S

_n

= (u

₁

+ u

_n

)+ (u

₁

+ u

_n

) + (u

₁

+ u

_n

) + .... + (u

₁

+ u

_n

) + (u

₁

+ u

_n

)

+ (u

₁

+ u

_n

)

n

times

2S

_n

= n(u

₁

+ u

_n

)

Where

So,

or

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(22)

Example:

Solution:

Find the sum of 4 + 7 + 10 + 13 + . . . To 50 term.

The series is arithmetic with u

₁

= 4, d = 3 and n = 50.

So,

For Senior High School

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For Senior High School

A geometric series is the addition of successive terms of a

geometric sequence.

A geometric series is the addition of successive terms of a

geometric sequence.

Recall that if the first term is u₁ and the common ratio is r, then the terms are:

So ,

Multiply S_n by r, the whole sequence get shifted along by one. We get :

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Factoring, then we get:

Or For

GEOMETRIC SERIES

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So, the sum of a geometric series is:

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EXAMPLE

:

Find the sum of 2 + 6 + 18 + 54 + ……….. to 12 terms!!

SOLUTION:

The series is geometric with u₁ = 2 , r = 3 and n = 12

So:

Using Then,

For Senior High School

(26)

Sometime it is necessary to consider when n gets

very large. What happens to S_n in this situation???

If i.e., then rn approaches 0 for every large

n

This mean that the series converges and has a sum to infinity of

So, the sum to infinity of geometric series is

GEOMETRIC SERIES

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(27)

EXAMPLE :

Find the sum of this infinite geometric series !!

SOLUTION:

The series is geometric with u₁ = 1, r = ½ So:

Using Then,

So, the sum of this infinite geometric series is 2

For Senior High School

(28)

For Senior High School

Motivation

Sequences Series

Refferences

Exit

Competence

(29)

FINAL TEST

DO YOUR BEST!!!

(30)

Final Test

Sequence and Series

Consider the sequence 6,17,28,39,50,…………

The general term and the 50th_{of this sequence is…………}

Problem 1

Home Back Next

A

B

B _DD

C

and 545 _{and 549}

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Problem 1

and 545 _{and 549}

and 535 and 542

CONGRATULATION!!!

TRUE!!

(32)

Problem 1

and 545 _{and 549}

and 535 and 542

DON’T GIVE UP!!!

FALSE!!

(33)

A sequence is defined by u_n = 3n -2, the least term of the sequence which is greater than 450 is………

Problem 2

A

B

B _DD

C

(34)

Problem 2

A

B

B _EE

C

CONGRATULATION!!!

TRUE!!

(35)

Problem 2

A

B

B _DD

C

DON’T GIVE UP!!!

FALSE!!

(36)

The initial population of rabbits on a farm was 50. the

population increased by 7% each week. How many rabbits when present after 30 weeks??? (using up rounding)

Problem 3

(37)

Problem 3

(38)

Problem 3

DON’T GIVE UP!!!

FALSE!!

(39)

A nest of ants initially consist of 500 ants. The population is increasing by 12% each weeks. After 10 weeks the population will be………

Problem 4

A

B

B _DD

C

1542 ants 1543 ants

1552 ants 1553 ants

(40)

1542 ants 1543 ants

1552 ants 1553 ants

CONGRATULATION!!!

TRUE!!

(41)

1542 ants 1543 ants

1552 ants 1553 ants

DON’T GIVE UP!!!

FALSE!!

(42)

The geometric sequence has u₂ = - 6 and u₅ = 162. the general term of that sequence is………

Problem 5

A

B

B _DD

C

(43)

Problem 5

A

B

B _DD

C

CONGRATULATION!!!

TRUE!!

(44)

Problem 5

A

B

B _DD

C

DON’T GIVE UP!!!

FALSE!!

(45)

The sum of 2 + 6 + 18 + 54 + ……… to 12 terms is………… Problem 6

Problem 6

A

B

B _DD

C

531.442

531.402 530.442

531.440

(46)

The sum of 2 + 6 + 18 + 54 + ……… to 12 terms is…………

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The sum of 2 + 6 + 18 + 54 + ……… to 12 terms is…………

DON’T GIVE UP!!!

FALSE!!

(48)

The general formula of 9 – 3 + 1 – 1/3 + ……… to n terms is………

Problem 7

A

B

B _DD

C

(49)

Problem 7

A

B

B _DD

C

CONGRATULATION!!!

TRUE!!

(50)

Problem 7

A

B

B _DD

C

DON’T GIVE UP!!!

FALSE!!

(51)

A ball takes 1 second to hit the ground when dropped. It then takes 90% of this time to rebound to its new height and this continues until the ball comes to rest. How long does it take for the ball to come to rest???

Problem 8

(52)

Problem 8

(53)

Problem 8

DON’T GIVE UP!!!

FALSE!!

(54)

The sum of -6 + 1 + 8 + 15 + …… + 141 is……… Problem 9

Problem 9

A

B

B _DD

C

1356

1565

1478

1485

(55)

The sum of -6 + 1 + 8 + 15 + …… + 141 is………

(56)

The sum of -6 + 1 + 8 + 15 + …… + 141 is………

DON’T GIVE UP!!!

FALSE!!

(57)

An arithmetic series has seven terms. The first term is 5 and the last term is 53. the sum of the series is…………

Problem 10

(58)

Problem 10

Final Test

Sequence and Series

(59)

Problem 10

DON’T GIVE UP!!!

FALSE!!

Final Test

Sequence and Series

(60)

THAT’S ALL FOR

TODAY

THAT’S ALL FOR

TODAY

T

HANKS

F

O

R

Y

OU

R

A

T

ENT

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