THE CONCEPT OF SEQUENCE AND SERIES. Adaptif Hal : 2 THE PROGRESSIONS Hal.: 2 The Pattern of Sequence...

THE CONCEPT OF SEQUENCE AND SERIES

Hal : 2 THE PROGRESSIONS AdaptifHal.: 2

The Pattern of Sequence and Series Number

Basic Competence:

Applying the concept of arithmetic sequence and

series

Indicator :1. The value of n-th term in an arithmetic sequence is defined

by formula

2. The sum of n in term of arithmetic sequence is defined by

formula


When you ride a motor cycle, have you ever look at the speeedometer?

In speedometer,there are numbers of 0,20, 40, 60, 80, 100, and 120 which show the speed of your motor cycle. These numbers are un order, starts from the smallest to the biggest with certain pattern, so that it forms a pattern of sequence



Imagine that you are a taxi passenger. You have to pay the starting fee Rp 15.000 and it charge Rp 2.500 /km.

15.000 17.500 20.000 22.500 …….

Starting fee 1 km 2 km 3 km 4 km



SIGMA NOTATION

The Concept of Sigma Notation

Look at the sum of the first sixth odd number below: 1 + 3 + 5 + 7 + 9 + 11 ……….. (1)

In the form(1) The 1st term = 1 = 2.1 – 1The 2nd term= 3 = 2.2 – 1The 3rd term = 5 = 2.3 – 1The 4th term = 7 = 2.4 – 1The 5th term = 9 = 2.5 – 1The 6th term = 11 = 2.6 – 1

Generally, the k-th term in (1) can be stated in the form of 2k – 1, k { 1, 2, 3, 4, 5, 6 }


SIGMA NOTATION

In Sigma notation, the addition form (1) can be written as:

6

1k

1)-(2k1197531


In the form of

6

1)12(

kk

It is read “sigma 2k – 1 from k =1 to 6” or “the sum

of 2k – 1 for k = 1 sd k = 6”

1 is called lower limit and

6 is called upper limit,

k is called index (some people

called it variable)

9

4)1)3(2(

kk

9

4)72(

kk

SIGMA NOTATION


SIGMA NOTATION

Generally


Stated into sigma form

1. a + a2b + a3b2 + a4b3 + … + a10b9

10

1k)1kbk(a

)142()132()122()112()12(4

1

k

k

Example:

249753

Define the value of

SIGMA NOTATION


SIGMA NOTATION

nnn

1n bCabC...baCbaCbaCa n1n

33nn3

22nn2

1nn1

n

n

0r

rrnnr baC

2. (a + b)n =


The properties of sigma notation :

, For every integer a, b and n

.....1 3211

n

n

k

aaaaak

n

mk

n

mk

akCCak.2

n

mk

n

mk

n

mk

bkakbkak )(.3

pn

pmk

n

mk

pakak.4

CmnCn

mk

)1(.5

n

mk

n

pk

p

mk

akakak1

.6

0.71

m

mk

ak

SIGMA NOTATION


SIGMA NOTATION

Example 1:

Show that

Answer :

3

1

3

1

)24()24(jk

ji

30)33.4()22.4()21.4()24(3

1

i

i

30)23.4()22.4()21.4()24(3

1

j

j


SIGMA NOTATION

6

4

23

1

2 66kk

kk

6

1

26

1

26

4

23

1

2 6666kkkk

kkkk

Define the value of

Example 2 :

Answer:

= 6 (12 +22 + 32 + 42 + 52 + 62)

= 6 (1 + 4 + 9 + 16 + 25 + 36)

= 6.91 = 546


ARITHMETIC SEQUENCE AND SERIES

The orderly numbers like in speedometer have the same difference for every two orderly term, so it forms a sequence

Arithmetic sequence is sequence with difference two orderly term constant

The general form is : U1, U2, U3, …., Un

a, a + b, a + 2b,…., a + (n-1)b

In arithmetic sequence, we have Un – Un-1 = b, so Un = Un-1 + b


If you start arithmetic sequence with the first term a and difference b, then you will get this following sequence

The n-th term of arithmetic sequence is Un = a + ( n – 1 )b

Where Un = n-th term

a = the first term

b = difference

n = the term’s quantity


a a + b a + 2b a + 3b …. a + (n-1)b


If every term of arithmetic sequence is added, then we will get arithmetic series.

Arithmetic series is the sum of terms of arithmetic sequence

General form :

U1 + U2 + U3 + … + Un atau

a + (a +b) + (a+2b) +… + (a+(n-1)b)

The formula of the sum of the first term in arithmetic series is

Where S = the sum of n-th term

n = the quantity of term

a = the first term

b = difference

= n-th term


bnan

Sn )1(22


Known: the sequence of 5, -2, -9, -16,…., find:

a.The formula of n-th term

b.The 25th term

Answer:

The difference of two orderly terms in sequence 5,-2, -9,-16 ,…is constant, b= -7,

so that the sequence is an arithmetic sequence

a.The formula of the n-th term in arithmetic sequence is

Un = 5 + ( n – 1 ). -7

Un = 5 + - 7n + 7

Un = -7n + 12

b. The 25th term of arithmetic sequence is : U12 = - 7.12 + 12

= - 163



Geometric sequence is a sequence which has the constant ratio between two orderly term

There is blue paper. It will cut into two pieces

GEOMETRIC SEQUENCE AND SERIES


Look at the paper part that form a sequence

Every two orderly terms of the sequence have the same ratio

It seems that the ratio of every two orderly terms in the sequence is always constant. The sequence like this is called geometric sequence and the comparison of every two orderly term is called ratio (r)

1 2 4

U1 U2 U3

2....12

3

1

2 n

n

U

U

U

U

U

U


Geometric sequence is a sequence which have constant ratio for two orderly term

General form: U1, U2, U3, …., Un atau

a, ar, ar2, …., arn-1

In geometric sequence

If you start the geometric sequence with the first term a and the ratio is r, then you get the following sequence


rU

U

n

n 1

1. nn UrsehinggaU


The n-th term of geometric sequence is :


Start With the first term a

Multiply with ratio r

Write the multiplication result



The relation of terms in geometric sequenceLike in arithmetic sequence, the relation between terms in geometric sequence can be explained as follows:

Take U12 as example :

U12 = a.r11

U12 = a.r9.r2 = U10. r2

U12 = a.r8.r3 = U9. r3

U12 = a.r4.r7 = U5. r7

U12 = a.r3.r8 = U4.r8

Generally, it can be formulated

Un = Uk. rn-k



Geometric series is the sum of terms in geometric sequenceGeneral form

U1 + U2 + U3 + …. + Un

a + ar + ar2 + ….+ arn-1

The formula of the n sum of the first term in geometric series is

1,1

)1(

rr

raS

n

n



Known sequence 27, 9, 3, 1, …..find

a.The formula of the n-th term

b. The 8th term

Answer:The ratio of two orderly terms in sequence 27,9,3, 1, …is constant,

so that the sequence is a geometric sequence

a. The formula of the n-th term in geometric sequence is

3

1r

1

3

127

n

nU

113 )3.(3 n

13 3.3 n

n 43

Hal : 25 THE PROGRESSIONS Adaptif


b. The 8th term of geometric sequence is

848 3 U

43

81

1

nnU

43


Infinite geometric series is a geometric series which has infinite terms.If infinite geometric series is -1 < r < 1 , then the sum of geometric series has sum limit (convergent).

For n = ∞ , rn is close to 0

So S∞ =

With S∞ = the sum of infinite geometric series a = the first term r = ratioIf r < -1 or r > 1 , then the infinite geometric series will be divergent, means the sum of terms is not limited

Infinite Geometric Series

r

a

1

r

raSn

n

1

)1(



1. Find the sum of infinite geometric series : 18 + 6 + 2 + … . .

Example :

3

1

2

3

1

2 u

u

u

ur


27

32

18

31

1

18

1

r

as

Answer :

a = 18 ;


2. An elastic ball is drop from the height of 2m. Every time it bounce from the floor, it has ¾ of the previous height. How long is the route that will be passed by the ball until it stop?


Look at the picture!The ball is drop from A, so AB is passed only once. Then CD, EF, etc is passed twice. The route is in geometric series with a = 3 and r = ¾ the length of the route is= 2 S∞ - a

2

412

2

2

43

1

22

12

a

r

a

= 14

So, the route length that pass by the ball is 14 m

THE CONCEPT OF SEQUENCE AND SERIES. Adaptif Hal : 2 THE PROGRESSIONS Hal.: 2 The Pattern of Sequence...

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Transcript of THE CONCEPT OF SEQUENCE AND SERIES. Adaptif Hal : 2 THE PROGRESSIONS Hal.: 2 The Pattern of Sequence...