1. Chapter 1
Section 1-2
An Application of Inductive Reasoning: Number Patterns
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2. An Application of Inductive Reasoning: Number Patterns
Number Sequences
Successive Differences
Number Patterns and Sum Formulas
Figurate Numbers
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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.
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4. Successive Differences
Process to determine the next term of a sequence using subtraction to find a common difference.
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5. Example: Successive Differences
Use the method of successive differences to find the next number in the sequence.
14, 22, 32, 44,...
Find differences
Find differences 58 14 2
Build up to next term: 58
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6. Number Patterns and Sum Formulas
Sum of the First n Odd Counting Numbers
If n is any counting number, then
Special Sum Formulas
For any counting number n,
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Example: Sum Formula
Use a sum formula to find the sum
Solution
Use the formula
with n = 48:
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Figurate Numbers
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9. Formulas for Triangular, Square, and Pentagonal Numbers
For any natural number n,
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10. Example: Figurate Numbers
Use a formula to find the sixth pentagonal number
Solution
Use the formula
with n = 6:
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Figurate Numbers
Hexagonal Numbers
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12. Example: Figurate Numbers
Use a formula to find the seventh hexagonal number
Solution
Use the formula
with n =7:
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Figurate Numbers
Triangular Numbers
Do you see the pattern? What would the formula be for Octagonal Numbers?
Square Numbers
Pentagonal Numbers
Hexagonal Numbers
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