? Finding the nth term of a Quadratic Sequence - Home Learning - Free Printable

1. For the sequence 8, 10, 14, 20, 28:
- First differences: 2, 4, 6, 8
- Second differences: 2, 2, 2 (constant)
- Since second differences are constant, it is quadratic.
- The coefficient of n² is half the second difference: 2 ÷ 2 = 1
- Subtract n² from original terms: (8-1), (10-4), (14-9), (20-16), (28-25) → 7, 6, 5, 4, 3
- This linear sequence has nth term: 8 - n
- Therefore, nth term of original sequence: n² - n + 8

2. For the sequence 1, 4, 9, 16, 25:
- First differences: 3, 5, 7, 9
- Second differences: 2, 2, 2 (constant)
- Coefficient of n²: 2 ÷ 2 = 1
- Subtract n²: (1-1), (4-4), (9-9), (16-16), (25-25) → 0, 0, 0, 0, 0
- Linear part is 0
- Therefore, nth term: n²

3. For the sequence 5, 10, 17, 26, 37:
- First differences: 5, 7, 9, 11
- Second differences: 2, 2, 2 (constant)
- Coefficient of n²: 2 ÷ 2 = 1
- Subtract n²: (5-1), (10-4), (17-9), (26-16), (37-25) → 4, 6, 8, 10, 12
- This linear sequence has nth term: 2n + 2
- Therefore, nth term of original sequence: n² + 2n + 2

4. For the sequence 10, 17, 26, 37, 50:
- First differences: 7, 9, 11, 13
- Second differences: 2, 2, 2 (constant)
- Coefficient of n²: 2 ÷ 2 = 1
- Subtract n²: (10-1), (17-4), (26-9), (37-16), (50-25) → 9, 13, 17, 21, 25
- This linear sequence has nth term: 4n + 5
- Therefore, nth term of original sequence: n² + 4n + 5

5. For the sequence 3, 10, 19, 30, 43:
- First differences: 7, 9, 11, 13
- Second differences: 2, 2, 2 (constant)
- Coefficient of n²: 2 ÷ 2 = 1
- Subtract n²: (3-1), (10-4), (19-9), (30-16), (43-25) → 2, 6, 10, 14, 18
- This linear sequence has nth term: 4n - 2
- Therefore, nth term of original sequence: n² + 4n - 2