Sequences

2.7 Sequences – Cheat Sheet

1. Continuing a Given Number Sequence or Pattern

• Identify the rule governing the sequence.
• Apply the rule to find missing terms.

2. Recognising Patterns in Sequences

Term-to-Term Rule:

• Describes how to get from one term to the next.
• Example: Start at 5, add 7 each time → 5, 12, 19, 26, …

Relationships Between Different Sequences:

• Example: Sequence B is always 3 more than Sequence A.
• Sequence A: 2, 4, 6, 8 → Sequence B: 5, 7, 9, 11

3. nth Term of a Linear Sequence

• Linear sequence: increases or decreases by a constant amount.
• nth term formula: Tₙ = an + b

Example:

• Sequence: 5, 8, 11, 14, …
• Common difference: +3 → a = 3
• First term: 5 → 3(1) + b = 5 → b = 2
• nth term: Tₙ = 3n + 2

4. nth Term of a Quadratic Sequence

• Second difference is constant.
• nth term formula: Tₙ = an² + bn + c
• Method:
  1. Find second difference → 2a = second difference → a = (second difference)/2
  2. Subtract an² from terms, find b and c using linear method.

Example:

• Sequence: 2, 6, 12, 20, …
• First difference: 4, 6, 8 → Second difference: 2 → 2a = 2 → a = 1
• Subtract n²: 1, 3, 5, 7 → Linear rule: 2n − 1 → b = 2, c = −1
• nth term: Tₙ = n² + 2n − 1

5. nth Term of a Cubic Sequence

• Third difference is constant.
• nth term formula: Tₙ = an³ + bn² + cn + d
• Method similar to quadratic but involves removing cubic and quadratic parts step-by-step.

6. Exponential Sequences

• Multiply by a constant each time.
• nth term formula: Tₙ = arⁿ⁻¹
  • a = first term, r = common ratio.

Example:

• Sequence: 3, 6, 12, 24, …
• a = 3, r = 2 → Tₙ = 3 × 2ⁿ⁻¹

7. Simple Combinations of Sequences

• Adding, subtracting, or multiplying sequences term by term.

Example:

• Sequence A: 1, 3, 5, 7, … (Tₙ = 2n − 1)
• Sequence B: 2, 4, 6, 8, … (Tₙ = 2n)
• Combined sequence (A + B): 3, 7, 11, 15, … (Tₙ = 4n − 1)

8. Examples Table for Quick Reference