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MATH 120
Concept of Numbers
Valerie Chu, Ph.D.
Summer 02
Chapter One
Section Two
Definitions:
1. Number Sequence:
A number sequence is a list of numbers having a first number, a second number, a third
number, and so on. Each number is a term.
2. Arithmetic Sequence:
An arithmetic sequence has a common difference between successive terms.
3. Geometric Sequence:
A geometric sequence has a common ratio between successive terms.
Examples:
1. The following is a sequence:
2, 6, 22, 56, 114, …….
The first term is 2, the second term is 6, and so on.
2. The following is an arithmetic sequence:
3, 9, 15, 21, 27, ……
What is the common difference between successive terms?
What will be the next term?
3. The following is a geometric sequence:
5, 10, 20, 40, 80, ……
What is the common ratio between successive terms?
What will be the next term?
The Method of Successive Differences:
If the next term of a sequence is not obvious, subtract the first term from the second
term, the second from the third, the third from the fourth, and so on. Repeat the process
with the new sequence and continue repeating until the difference is a constant value.
Once a line of constant values is obtained, simply work “backward” by adding until the
desired term of the given sequence is obtained.
Example: (Let us use Microsoft Excel to do the jobs!)
2
6
22
56
4
16
34
12
18
24
6
6
Try the following sequences:
14
22
5
15
32
44
37
77
1
114
58
202
88
30
6
141
Exercises: (Page 18)
Use the method of successive differences to determine the next number in each sequence.
The answers are
1. 79
3. 450
5. 4032
Number Patterns:
• Sum of the first n counting numbers:
Can you find the sum of 1 + 2 + 3 + 4 + ……. +60?
Try this:
S=1
+2
+3
+4
S = 60
+59
+58
+57
2S= 61
+61
+61
+61
That is,
60 × 61
2 S = 60 × 61
S=
2
Thus, we get the formulas as
1 + 2 + 3 + 4 + .... + n =
7. 32,758
+
+
+
S=
+59
+2
+61
60 × (60 + 1)
2
n(n + 1)
2
Exercises: (Page 18)
Find each of the following sums:
1. 1 + 2 + 3 + 4 + ……. +300=____________
Answers
45,150
2. 1 + 2 + 3 + 4 + ……. +675=____________
228,150
•
Sum of the first n odd counting numbers:
Therefore, if you follow the pattern, you can get
Thus, the formulas is
1 + 3 + 5 + 7 + ....+ (2n −1) = n2
Exercises: (Page 18)
Find each of the following sums:
1. 1 + 3 + 5 + 7 + ……. +101=____________
Answers
2601
2. 1 + 3 + 5 + 7 + ……. +999=____________
250,000
2
+60
+1
+61
Page 19:
•
The square of the sum of first n counting numbers and sum of cube of each number:
If keep going, you will get
Thus, the formulas is
(1 + 2 + 3 + 4 + .....+ n) 2 = 13 + 23 + 33 + 43 + .....+ n3
Exercise:
Compute the sum of the following expression:
1. 1 3 + 2 3 + 3 3 + ...... + 20 3 = ?
20 × (20 + 1) 20 × 21
First, compute 1 + 2 + 3 + .... + 20 =
= 10 × 21 = 210
=
2
2
Then, square the result. 210 2 = 44,100
Therefore, 1 3 + 2 3 + 3 3 + ...... + 20 3 = 44 ,100
2. 1 3 + 2 3 + 3 3 + ...... + 100 3 = ?
Try this by yourself.
•
Sum of Odd Counting Numbers: (Page 14)
Exercise: 31 + 33 + 35 + 37 + 39 +41 = ?
3
•
Figurate Numbers:
Can you find the pattern of the above formula?
4