4.2.3

Types of Sequences

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Types of Sequences

We can determine whether a sequence is increasing, decreasing or periodic by looking at patterns in the terms of the sequence.

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Increasing sequence

  • A sequence is increasing if each term of the sequence is greater than the previous term.
  • We can write this condition as:
    • un+1>un for all nu_{n+1}>u_n \text{ for all } n\in
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Example

  • These sequences are all increasing sequences:
    • 1,2,3,4,...1, 2, 3, 4,...
    • 1,4,9,16,...1, 4, 9, 16, ...
    • 6,12,24,48,...6, 12, 24, 48, ...
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Decreasing sequences

  • A sequence is decreasing if each term of the sequence is less than the previous term.
  • We can write this condition as:
    • un+1<un for all nu_{n+1}<u_n\text{ for all }n\in
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Examples

  • These sequences are all decreasing sequences:
    • 4,5,6,7,...-4,-5,-6,-7,...
    • 7,3,2,0,...7, 3, 2, 0, ...
    • 100,98,96,94,...100, 98, 96, 94, ...
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Periodic

  • A sequence is periodic if the terms of the sequence repeat after a certain number of terms.
  • If a sequence is periodic, then there is an integer kk such that:
    • un+k=un for all nu_{n+k} = u_n\text{ for all }n \in
  • Where kk is called the order of the sequence.
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Examples

  • The sequence 0,1,0,1,0,1,0,1,....0, 1, 0, -1, 0, 1, 0, -1, .... is a periodic sequence of order 4, as the nnth term is the same as the n+4n+4th term.
  • The sequence 5,8,5,8,5,8,...5,8,5,8,5,8,... is a periodic sequence of order 2, as the nnth term is the same as the n+2n+2th term.
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None of the above

  • A sequence can have terms that are not increasing, decreasing or periodic.
  • For example, the sequence 10,10,8,8,7,7,....10, -10, 8, -8, 7, -7, .... is not increasing as each term is not greater than the previous term.
  • It is not decreasing because each term is not less than the previous term.
  • It is not periodic because there is no repetition of terms.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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