Chapter 1 sequences and series
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Transcript of Chapter 1 sequences and series
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)