Study Notes
In this topic, you will learn how to describe the rule for continuing a sequence and find the nth term of sequences, including linear, quadratic, cubic, and geometric sequences. You will also learn to generate sequences from patterns of shapes.
- Linear Sequence — a sequence where the difference between consecutive terms is constant. Example: 6, 10, 14, 18 has a first difference of +4, so the nth term is 4n - 2.
- Quadratic Sequence — a sequence where the second difference between terms is constant. Example: 2, 7, 14, 23, 34 has a second difference of 2, so the nth term is n² + 2n - 1.
- Cubic Sequence — a sequence where the third difference between terms is constant. Example: 4, 16, 44, 94, 172, 284 has a third difference of 6, so the nth term is n³ + 2n² - n + 2.
- Geometric Sequence — a sequence where each term is obtained by multiplying the previous term by a constant. Example: A sequence with first term a and common ratio r.
Exam Tips
Key Definitions to Remember
- Linear sequence: constant first difference
- Quadratic sequence: constant second difference
- Cubic sequence: constant third difference
- Geometric sequence: constant ratio between terms
Common Confusions
- Confusing the difference method for linear, quadratic, and cubic sequences
- Mixing up the terms 'sequence' and 'series'
Typical Exam Questions
- What is the nth term of the sequence 6, 10, 14, 18? The nth term is 4n - 2.
- How do you find the nth term of a quadratic sequence? Use the second difference to find a, then solve for b and c.
- What is a geometric sequence? A sequence where each term is multiplied by a constant ratio.
What Examiners Usually Test
- Ability to find the nth term of various sequences
- Understanding of the difference method for sequences
- Application of sequences to solve problems