Download An Application of Inductive Reasoning: Number Patterns

Chapter 1
The Art of
Problem
Solving
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All rights reserved
Chapter 1: The Art of Problem Solving
1.1 Solving Problems by Inductive
Reasoning
1.2 An Application of Inductive Reasoning:
Number Patterns
1.3 Strategies for Problem Solving
1.4 Calculating, Estimating, and Reading
Graphs
1-2-2
© 2008 Pearson Addison-Wesley. All rights reserved
Chapter 1
Section 1-2
An Application of Inductive
Reasoning: Number Patterns
1-2-3
© 2008 Pearson Addison-Wesley. All rights reserved
An Application of Inductive
Reasoning: Number Patterns
•
•
•
•
Number Sequences
Successive Differences
Number Patterns and Sum Formulas
Figurate Numbers
1-2-4
© 2008 Pearson Addison-Wesley. All rights reserved
Number Sequences
Number Sequence
A list of numbers having a first number, a second
number, and so on, called the terms of the sequence.
Arithmetic Sequence
A sequence that has a common difference between
successive terms.
Geometric Sequence
A sequence that has a common ratio between successive
terms.
1-2-5
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Successive Differences
Process to determine the next term of a sequence
using subtraction to find a common difference.
1-2-6
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Successive Differences
Use the method of successive differences to find the
next number in the sequence.
14, 22, 32, 44,...
14
22
8
32
10
2
44
12
2
58
14
2
Find differences
Find differences
Build up to next term: 58
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Number Patterns and Sum
Formulas
Sum of the First n Odd Counting Numbers
If n is any counting number, then
1 3  5 
 (2n  1)  n 2 .
Special Sum Formulas
For any counting number n,
(1  2  3 
 n)2  13  23   n3
n(n  1)
and 1  2  3   n 
.
2
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Example: Sum Formula
Use a sum formula to find the sum
1 2  3 
 48.
Solution
Use the formula 1  2  3 
with n = 48:
n(n  1)
n
2
48(48  1)
 1176.
2
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Figurate Numbers
1-2-10
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Formulas for Triangular, Square,
and Pentagonal Numbers
For any natural number n,
n(n  1)
the nth triangular number is given by Tn 
,
2
the nth square number is given by Sn  n , and
2
n(3n  1)
the nth pentagonal number is given by Pn 
.
2
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Example: Figurate Numbers
Use a formula to find the sixth pentagonal
number
Solution
n(3n  1)
Use the formula Pn 
2
with n = 6:
6[6(3)  1]
P6 
 51.
2
1-2-12
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