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Section 1-2

Section 1-2. An Application of Inductive Reasoning: Number Patterns. An Application of Inductive Reasoning: Number Patterns. Number Sequences Successive Differences Number Patterns and Sum Formulas Figurate Numbers. Number Sequences. Number Sequence

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Section 1-2

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  1. Section 1-2 • An Application of Inductive Reasoning: Number Patterns

  2. An Application of Inductive Reasoning: Number Patterns • Number Sequences • Successive Differences • Number Patterns and Sum Formulas • Figurate Numbers

  3. 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.

  4. Successive Differences Process to determine the next term of a sequence using subtraction to find a common difference.

  5. Example: Successive Differences Use the method of successive differences to find the next number in the sequence. 14, 22, 32, 44,... 14 22 32 44 8 10 12 Find differences 2 2 Find differences 58 14 2 Build up to next term: 58

  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,

  7. Example: Sum Formula Use a sum formula to find the sum Solution Use the formula with n = 48:

  8. Figurate Numbers

  9. Formulas for Triangular, Square, and Pentagonal Numbers For any natural number n,

  10. Example: Figurate Numbers Use a formula to find the sixth pentagonal number Solution Use the formula with n = 6:

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