Finding the Nth Term of a Quadratic Sequence

Introduction

First Difference: Difference between consecutive terms
- Example:
  - 6 to 19: add 13
  - 19 to 38: add 19
  - 38 to 63: add 25
  - 63 to 94: add 31
- This is called the first difference.
- In linear sequences, the first difference is constant.
Second Difference: Difference between first differences
- Example:
  - 13 to 19: add 6
  - 19 to 25: add 6
  - 25 to 31: add 6
- This is called the second difference.
- If the second difference is constant, the sequence is quadratic.

Coefficient of n²
- The coefficient of n² is half of the second difference.
- For a second difference of 6, the coefficient of n² is 3.
Building the Quadratic Sequence
- Start with known term, e.g., 3n².
- Generate terms for 3n²:
  - 1, 4, 9, 16, 25 times 3: 3, 12, 27, 48, 75.
- Compare with the original sequence:
  - Original: 6, 19, 38, 63, 94.
  - 3n²: 3, 12, 27, 48, 75.
- Subtract to find the linear part:
  - Example: 6-3=3, 19-12=7, etc.
- This results in a linear sequence: 3, 7, 11, 15, 19.
- Find the nth term for linear sequence:
  - Coefficient of n: 4.
  - Sequence doesn't start at 4, it starts at 3, so adjust: 4n-1.
Combining Both Parts
- Add quadratic part (3n²) and linear part (4n-1) together.
- Final nth term: 3n² + 4n - 1.

First Differences
- 4 to 7: add 3
- 7 to 14: add 7
- 14 to 25: add 11
- 25 to 40: add 15
Second Differences
- 3 to 7: add 4
- 7 to 11: add 4
- 11 to 15: add 4
- Second difference is 4 (constant), so quadratic sequence.
Finding nth term
- Second difference: 4, half is 2, so coefficient of n² is 2.
- Sequence starts with 2n².
- Generate 2n² sequence:
  - 1, 4, 9, 16, 25 times 2: 2, 8, 18, 32, 50.
- Subtract from original sequence:
  - Example: 4-2=2, 7-8=-1, etc.
- Linear sequence: 2, -1, -4, -7, -10.
- Find nth term for linear sequence:
  - Coefficient of n: -3.
  - Sequence starts at 2, not -3, so adjust: -3n + 5.
- Combine parts:
  - Final nth term: 2n² - 3n + 5.