CHAPTER 1

SEQUENCES AND SERIES

MALAYSIA INSTITUTE OF INFORMATION TECHNOLOGYUNIVERSITI KUALA LUMPUR

UPDATED: JUN 2011

1.1 SEQUENCE

Consider the following sequence:

Sequence 1 Sequence 2

2, 4, 6, 8, 10 2, 4, 6, 8, 10…..

1st term2nd term3rd term

4th term5 th term

1st term2nd term3rd term

4th term5 th term

Each no in the list is called a term.

Each term are separated by commas.

Finite sequence end after a certain no of terms.

Infinite sequence is one that continues indefinitely (non stop).

finite sequence infinite sequence

1.1.1 Finding the formula for the nth terms of a sequence

Example 1:

Find the nth term of the following sequence

Solution: Solution:

The seq. can be written as The seq. can be written as

The nth term is The nth term is

,...,,,a625

2

125

2

25

2

5

2

The formula for the terms in sequence can be given as a formula for the nth term.

.....,,,, 142132122112 ,...,,,4321 5

2

5

2

5

2

5

2

2/(52/(5n))

,...,,, b 9753

2(n)+12(n)+1

Solution:

The seq. can be written as

The nth term is

Example 2: Example 3:

Write the first 6 terms of x(k)=5-k.

Solution: X(1)=5-1=4 X(2)=5-2=3 X(3)=5-3=2 X(4)=5-4=1 X(5)=5-5=0 X(6)=5-6=-1

The first 6 terms is

4,3,2,1,0,-1

Write the first 6 terms of x(k)=2k.

Solution:

The first 6 terms is

2,4,8,16,32,64

,....,,,c 201262

1.2 SERIESConsider the following sequence:

Series 1 Series 2

2 + 4 + 6 + 8 +10 2 + 4 + 6 + 8 + 10…..

Example:

Finite series

Infinite series

222225

1

2 53433323133 i

i

1st term 2nd term3rd term

4th term5 th term

1st term2nd term3rd term

4th term5 th term

Series is the sum of terms in a sequence.

Summation notation is

Finite series end after a certain no of terms.

Infinite series is one that continues indefinitely (non stop).

finite series infinite series

n

rru

Last value of r in the sequence

First value of r in the sequence

...ii

22222

1

2 53433323133

First value of r in the sequence

Infinity

Example 4:

Write the terms of the series.

Solution: Solution:

Example 5:

Write in the summation notation

Solution: Solution:

The series can be written as: The series can be written as:

5

2

2ma

2222 5432

k

mmb1

1

134231201 kk....

1001284 ...a

254342414 ....

25

1

4i

i

....b 5

4

4

3

3

2

2

1

....

41

4

31

3

21

2

11

1

1 1i i

i

Example 6:

Determine the sum of the following series.

Solution:

Solution:

110

5

2 ka

361

1018265503726

1101918171615 222222

20

113

20

716

5

5

4

4

3

3

2

2

1

15

5

14

4

13

3

12

2

11

1

OR

5

1 1n

nb

1. 3 ARITHMETIC SEQUENCE AND SERIES1.3.1 ARITHMETIC SEQUENCE

The difference between consecutive terms is same. The difference is called common difference, d.

Consider the following sequence

Each terms has 4 added to it to obtain the next term. The sequence is said to have a common difference of 4 (d = 4).

....,,,, 141062

14 -106 -2 10 - 6d = 4 d = 4 d = 4

Formula for the nth term, a = the first term d = common difference

Example 7:

Find the tenth term of the arithmetic sequence: 32, 47, 62, 77,…..

167

1511032T10

dna 1Tn

dna 1Tn

15

32 - 47d

32a

Example 8:

Find the number of terms in the arithmetic sequence: 50, 47, 44, …., -34

293

87

343503

315034-

n

n

n

dna 1Tn

3

50 - 47d

50a

1.3.2 ARITHMETIC SERIES

A series is the sum of a sequenceFormula for sum of an arithmetic series,

a = the first term d = common difference l = the last term

Example 9:

Determine the sum of the first 25 terms of arithmetic series:

..... 14161820

dnan

S

ORlan

S

n

n

122

2

25220 n,d,a

100

21252022

2525

S

Example 10:

52 d,a

771272:seriesarithmeticofsumtheDetermine .....

165

80

52775

55277

51277

1

n

n

n

n

dnaTn

632

7722

1616

S

Exercises

1. Find the 23rd term of an arithmetic sequence with first term 2 and common difference 7. (156)

2. Find the sum of the first five terms of the arithmetic sequence with first term 3 and common difference 5. (65)

3. Write down the 10th and 19th terms of the arithmetic sequence

a. 8, 11, 14, … (35, 62)

b. 8, 5, 2, … (-19, -46)

4. An arithmetic sequence is given by

a. State the sixth term(-2b/3)

b. State the kth term [b(4-k)]/3

c. If the 20th term has a value of 15, find b. -45/16,...0 ,

3 ,

3

2 ,

bbb

Application Problems1. After knee surgery, your trainer tells you to

return to your jogging program slowly.  He suggests jogging for 12 minutes each day for the first week.  Each week thereafter, he suggests that you increase that time by 6 minutes per day.  How many weeks will it be before you are up to jogging 60 minutes per day?

2. You visit the Grand Canyon and drop a penny off the edge of a cliff.  The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence.  What is the total distance the object will fall in 6 seconds?

Application Problems3. A shop assistant is arranging a display of a

triangular array of tins so as to have one tin in the top row, two in the second, three in the third and so on. If there are 66 tins altogether, how many rows can be completed arranged? (11)

4. A restaurant has square tables which seat four people. When two tables are placed together, six people can be seated.

If 20 square tables are placed together to form one long table, how many people can be seated?

If 1000 square tables are placed together to form one very long table, how many people can be seated?(42, 2002)

1. 4 GEOMETRIC SEQUENCE AND SERIES1.4.1 GEOMETRIC SEQUENCE

The ratio between consecutive terms is same. The ratio is called common ratio, r.

Consider the following sequence

Starting with 4, and multiplying each term by -2 would generate the above sequence.

The sequence is said to have a common ratio of -2 (r = -2).

....,,,, 321684

-8/4 16/-8 -32/16r = -2 r = -2 r = -2

Formula for the nth term, a = the first term r = common ratio

Example 11:

Determine the eighth term of the geometric sequence: 4, 12, 36,108,…..

1nT nar

34

12r

8748

34T 7

8

1nT nar

4a

1.4.2 GEOMETRIC SERIES

A series is the sum of a sequenceFormula for the sum of a geometric series,

a = the first term r = common ratio

Example 12:

Determine the sum of the first 10 terms of geometric series:

11

1

11

1

rifr

raS

rifr

raS

n

n

n

n

1

102

14

r

n,r,a

.....2

1124

128

1023

21

1

21

14

10

10

S

Example 13:

1

23

r

r,a.....terms 20 first theofne 241263:seriesgeometricofsumtheDetermi

7251453

12

123

1

1

20

20

S

r

raS

n

n

Formula for the sum of a Infinite geometric series,

a = the first term r = common ratio

Example 12:

Determine the sum of a Infinite geometric series with the first term is 3 and the common ration is ½ :

111

rifr

aS

6213

21

1

3

S

ExercisesExercises1. Find the seventh term of a geometric sequence with first

term 2 and common ratio 3. (1458)

2. Find the sum of the first five terms of the geometric sequence with the first term 3 and common ratio 2. (93)

3. Find the sum of the infinite geometric series with first term 2 and common ratio ½. (4)

4. A geometric sequence has first term 1. The ninth term exceed the fifth term by 240. Find possible values for the eighth term. (+/- 128)

5. The sum to infinity of a geometric sequence is four times the first term. Find the common ratio. (3/4)

6. A geometric series has S3 = 37/8 and S6 = 367/512. Find the first term and the common ratio. (2, ¾)

7. A geometric sequence is given by 1, ½, ¼, … What is its common ratio? (1/2)

Solution Exercise No 4Solution Exercise No 4

4. A geometric sequence has first term 1. The ninth term exceed the fifth term by 240. Find possible values for the eighth term. (+/- 128)

1282;2if

1282;2if

216Since

15;16

01516

0240

0240;Let

240

.1;.1;240

1

78

78

4

2

42.44

48

4155

819959

ar

ar

rx

xx

xx

xx

rrrx

rr

rrarraaa

aStep 1:

Step 2:

Step 3:

Solution Exercise No 6Solution Exercise No 66. A geometric series has S3 = 37/8 and S6 = 3367/512.

Find the first term and the common ratio. (2, ¾ )

2;Eqintosubs18

37

1;1From

2Eq512

3367

1

1

1Eq8

37

1

1

1assume

;512

3367;

8

37

3

6

6

3

3

63

rr

a

r

raS

r

raS

r

SS

75.0@4

3

64

27

64

2764

911

37

8

512

3367

1

11

512

3367

18

371

3

3

3

3

33

36

r

r

r

r

rr

rr

Step 1:

Step 2:

Step 3:

2

18

137

18

137

837

437

6427

43

34

3

43

aStep 4:

Application problems

1. A culture of bacteria doubles every 2 hours.  If there are 500 bacteria at the beginning, how many bacteria will there be after 24 hours?

2. Recent estimates, based on data from satellite observations, report 775 million hectares of rain forest remaining. The average annual rate of deforestation in the world is 0.77%. How many million hectares of rain forest will be lost in the next decade? (57.65 millions)