Overview
This lecture explains how to find the nth term of a quadratic sequence by analyzing differences and constructing a formula, building on linear sequence skills.
Identifying Quadratic Sequences
- Find first differences between sequence terms; if not constant, check for quadratic sequence.
- Find second differences between first differences; if constant, the sequence is quadratic.
Finding the nth Term Formula
- The coefficient of n² in the nth term is half the constant second difference.
- Start forming the nth term as (second difference ÷ 2) × n².
- Generate the sequence for this quadratic part (e.g., 3n²) and compare to the original sequence.
Determining the Linear Component
- Subtract the quadratic sequence from the original to get a new (linear) sequence.
- Find the nth term for the new linear sequence (usually of the form an + b).
- Add the quadratic and linear nth terms together for the full formula.
Examples
- Example 1: For sequence 6, 19, 38, 63, 94:
- Second difference is 6, so start with 3n².
- Subtracted sequence gives differences with 4n – 1, so nth term: 3n² + 4n – 1.
- Example 2: For sequence 4, 7, 14, 25, 40:
- Second difference is 4, so start with 2n².
- Subtracted sequence gives –3n + 5, so nth term: 2n² – 3n + 5.
Key Terms & Definitions
- Quadratic Sequence — a sequence where the second differences are constant.
- First Difference — the difference between consecutive terms in a sequence.
- Second Difference — the difference between consecutive first differences.
Action Items / Next Steps
- Practice finding nth terms for quadratic sequences using the method shown.
- Review how to find the nth term of linear sequences if needed.