Polynomial Sequences & Method of Differences (a·n² + b·n + c)
Core Pattern: When 1st differences are not constant, take the differences of the differences (the 2nd differences). For any quadratic sequence, the 2nd differences are CONSTANT ($2a$). Extend the 2nd difference line to deduce the next number.
Mathematical Formulation & Recurrence
\(x_n = a n^2 + b n + c \iff \Delta^{(2)} = 2a \quad \text{(Constant 2nd Difference)}\)
Method of Differences & Calculation Technique
The 2nd Difference Table Method (Universal Additive Engine):
1. Row 0 (Terms): Write out the sequence: \(x_1, x_2, x_3, x_4, x_5\).
2. Row 1 (1st Differences \(\Delta^{(1)}\)): Subtract adjacent terms \(\Delta^{(1)}_k = x_{k+1} - x_k\). Notice that this row is NOT constant, but increases by a steady step!
3. Row 2 (2nd Differences \(\Delta^{(2)}\)): Subtract adjacent 1st differences \(\Delta^{(2)}_k = \Delta^{(1)}_{k+1} - \Delta^{(1)}_k\). This row IS CONSTANT! For \(a n^2 + b n + c\), this constant is exactly \(2a\).
4. Reconstruct Next Number (Bottom-Up): Extend Row 2 with the constant \(K\). Add \(K\) to the last 1st difference: \(\Delta^{(1)}_{\text{next}} = \Delta^{(1)}_{\text{last}} + K\). Add \(\Delta^{(1)}_{\text{next}}\) to the last term: \(x_{\text{next}} = x_{\text{last}} + \Delta^{(1)}_{\text{next}}\).
Visual Sequence & Difference Mapping
How to Solve This Series in 3 Steps:
- 1 Compute Row 1 (1st Differences): Subtract each term from the one after it.
- 2 Compute Row 2 (2nd Differences): Subtract each 1st difference from the one after it to find the constant value K = 2a.
- 3 Extend Row 2: Write down K again for the next gap.
- 4 Step up to 1st Differences: Add K to the last 1st difference to get the new 1st difference.
- 5 Step up to Terms: Add that new 1st difference to the last term in the sequence to get the answer!
Step-by-Step Worked Examples:
2, 7, 16, 29, 46, ?
Example 1
1. 1st differences: 7-2=5, 16-7=9, 29-16=13, 46-29=17
2. 2nd differences: 9-5=4, 13-9=4, 17-13=4 (Constant Δ² = 4)
3. Next 1st diff: 17 + 4 = 21. Next term: 46 + 21 = 67
Answer: 67
1, 4, 9, 16, 25, ? (n² squares)
Example 2
1. 1st differences: 3, 5, 7, 9 (odd numbers)
2. 2nd differences: 2, 2, 2 (Constant Δ² = 2)
3. Next 1st diff: 9 + 2 = 11. Next term: 25 + 11 = 36
Answer: 36
3, 9, 19, 33, 51, ? (2n² + 1)
Example 3
1. 1st differences: 6, 10, 14, 18
2. 2nd differences: 4, 4, 4 (Constant Δ² = 4 → a = 2)
3. Next 1st diff: 18 + 4 = 22. Next term: 51 + 22 = 73
Answer: 73
0, 5, 14, 27, 44, ? (2n² - n - 1)
Example 4
1. 1st differences: 5, 9, 13, 17
2. 2nd differences: 4, 4, 4 (Constant Δ² = 4)
3. Next 1st diff: 17 + 4 = 21. Next term: 44 + 21 = 65
Answer: 65
Interactive Comprehension Test
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3, 8, 15, 24, 35, ?
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