Sequences Revision Notes

A number sequence is an ordered list of numbers following a specific pattern or rule. To extend a sequence, examine the differences or ratios between consecutive terms to identify the pattern.

For example, consider the sequence: 1, 3, 6, 10, 15, … To find the next terms, calculate the differences between terms:

3 - 1 = 2, 6 - 3 = 3, 10 - 6 = 4, 15 - 10 = 5. These differences increase by 1 each time, so the next differences are 6 and 7. Adding these to the last term: 15 + 6 = 21, then 21 + 7 = 28. Thus, the next two terms are 21 and 28.

• Calculate differences between consecutive terms.
• Look for constant or regularly changing differences.
• Use the pattern in differences to find next terms.

Sequences may be linear, quadratic, cubic, or follow other patterns. Understanding the relationship between terms helps to find a general formula (nth term) that generates any term in the sequence.

Linear sequences have a constant difference between terms. Quadratic sequences have differences of differences constant (second differences). Cubic sequences have constant third differences.

Recognising these patterns allows you to classify the sequence and apply the correct method to find the nth term.

• Linear sequences: constant first difference.
• Quadratic sequences: constant second difference.
• Cubic sequences: constant third difference.
• Use differences to classify the sequence type.

A linear sequence increases or decreases by a constant amount each time. The nth term formula is: an = a + (n - 1)d, where:

a is the first term, d is the common difference, n is the term number.

For example, for the sequence 4, 7, 10, 13, …, the first term a = 4 and the common difference d = 3. The nth term is: an = 4 + (n - 1) × 3 = 3n + 1. To find the 10th term: an = 3(10) + 1 = 31.

• Identify first term and common difference.
• Substitute into an = a + (n - 1)d.
• Calculate specific terms by substituting n.

Quadratic sequences have a second difference that is constant. The nth term is generally of the form an = an² + bn + c, where a, b, and c are constants.

To find a, b, and c: calculate first and second differences, use simultaneous equations from the first few terms, then solve for a, b, and c.

Example: For the sequence 2, 5, 10, 17, …, the first differences are 3, 5, 7 and the second differences are constant at 2. Since the second difference = 2a, then 2a = 2 so a = 1. Using the first term (n=1): a(1)² + b(1) + c = 2 → 1 + b + c = 2 → b + c = 1. Using the second term (n=2): 4a + 2b + c = 5 → 4 + 2b + c = 5 → 2b + c = 1. Solving these simultaneously gives b = 0 and c = 1. Thus, nth term is an = n² + 1.

• Calculate first and second differences.
• Use second difference to find a.
• Form simultaneous equations using terms.
• Solve for b and c.
• Write nth term as an = an² + bn + c.

Cubic sequences have a constant third difference. The nth term is generally of the form an = an³ + bn² + cn + d, where a, b, c, and d are constants.

Finding the nth term involves calculating first, second, and third differences, then using these to set up equations to solve for the constants.

Example: For the sequence 1, 8, 27, 64, 125, … (cubes of natural numbers), the nth term is an = n³. The third difference is constant, confirming the cubic pattern.

• Calculate first, second, and third differences.
• Use third difference to find a.
• Set up equations using initial terms.
• Solve for b, c, and d.
• Write nth term as an = an³ + bn² + cn + d.