4.2.3
Types of Sequences
Types of Sequences
Types of Sequences
We can determine whether a sequence is increasing, decreasing or periodic by looking at patterns in the terms of the sequence.
![Illustrative background for Increasing sequence](https://image-v2.cdn.app.senecalearning.com/2020-06/132919a4-ca15-4b34-9819-6a55cb89c721/Screenshot%202020-06-16%20at%2011.36.21-min,h_400,q_80,w_640.png)
![Illustrative background for Increasing sequence ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/132919a4-ca15-4b34-9819-6a55cb89c721/Screenshot%202020-06-16%20at%2011.36.21-min,h_400,q_80,w_640.png)
Increasing sequence
Increasing sequence
- A sequence is increasing if each term of the sequence is greater than the previous term.
- We can write this condition as:
- ℕ
![Illustrative background for Example](https://image-v2.cdn.app.senecalearning.com/2020-06/1cedaa2a-9e99-4b3a-9f28-3ccfd3e9907e/Screenshot%202020-06-16%20at%2011.36.23-min,h_400,q_80,w_640.png)
![Illustrative background for Example ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/1cedaa2a-9e99-4b3a-9f28-3ccfd3e9907e/Screenshot%202020-06-16%20at%2011.36.23-min,h_400,q_80,w_640.png)
Example
Example
- These sequences are all increasing sequences:
![Illustrative background for Decreasing sequences](https://image-v2.cdn.app.senecalearning.com/2020-06/0d2d5bbd-4b13-4fb1-baf2-65514b74a214/Screenshot%202020-06-16%20at%2011.36.24-min,h_400,q_80,w_640.png)
![Illustrative background for Decreasing sequences ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/0d2d5bbd-4b13-4fb1-baf2-65514b74a214/Screenshot%202020-06-16%20at%2011.36.24-min,h_400,q_80,w_640.png)
Decreasing sequences
Decreasing sequences
- A sequence is decreasing if each term of the sequence is less than the previous term.
- We can write this condition as:
- ℕ
![Illustrative background for Examples](https://image-v2.cdn.app.senecalearning.com/2020-06/97936068-1061-4668-8369-60f933f53ab0/Screenshot%202020-06-16%20at%2011.36.26-min,h_400,q_80,w_640.png)
![Illustrative background for Examples ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/97936068-1061-4668-8369-60f933f53ab0/Screenshot%202020-06-16%20at%2011.36.26-min,h_400,q_80,w_640.png)
Examples
Examples
- These sequences are all decreasing sequences:
![Illustrative background for Periodic](https://image-v2.cdn.app.senecalearning.com/2020-06/4f22cbc9-92ea-457b-ab63-9c09b4c9bfb4/Screenshot%202020-06-16%20at%2011.36.37-min,h_400,q_80,w_640.png)
![Illustrative background for Periodic ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/4f22cbc9-92ea-457b-ab63-9c09b4c9bfb4/Screenshot%202020-06-16%20at%2011.36.37-min,h_400,q_80,w_640.png)
Periodic
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 such that:
- ℕ
- Where is called the order of the sequence.
![Illustrative background for Examples](https://image-v2.cdn.app.senecalearning.com/2020-06/b4c99035-3979-4818-b8fd-240e3ba3a670/Screenshot%202020-06-16%20at%2011.41.24-min,h_400,q_80,w_640.png)
![Illustrative background for Examples ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/b4c99035-3979-4818-b8fd-240e3ba3a670/Screenshot%202020-06-16%20at%2011.41.24-min,h_400,q_80,w_640.png)
Examples
Examples
- The sequence is a periodic sequence of order 4, as the th term is the same as the th term.
- The sequence is a periodic sequence of order 2, as the th term is the same as the th term.
![Illustrative background for None of the above](https://image-v2.cdn.app.senecalearning.com/2020-06/be1abf75-a0ce-4d54-aa9f-9ab0b40a53af/Screenshot%202020-06-16%20at%2011.36.32-min,h_400,q_80,w_640.png)
![Illustrative background for None of the above ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/be1abf75-a0ce-4d54-aa9f-9ab0b40a53af/Screenshot%202020-06-16%20at%2011.36.32-min,h_400,q_80,w_640.png)
None of the above
None of the above
- A sequence can have terms that are not increasing, decreasing or periodic.
- For example, the sequence 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.
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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