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Difference Method
A systematic technique for identifying sequence patterns by calculating differences between consecutive terms.
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A systematic technique for identifying sequence patterns by calculating differences between consecutive terms.
The difference method is a fundamental problem-solving technique for number sequences. By calculating the differences between consecutive terms, you can often reveal the underlying pattern. If the first differences are constant, it's arithmetic. If the first differences vary but the second differences are constant, it's quadratic. This method systematically uncovers hidden structure in sequences.
First differences: dₙ = aₙ₊₁ - aₙ
- Sequence: 2, 5, 8, 11 → Differences: 3, 3, 3 (constant = arithmetic)
- Sequence: 1, 4, 9, 16 → Differences: 3, 5, 7 → Second differences: 2, 2 (quadratic)
- Sequence: 2, 6, 18, 54 → Differences: 4, 12, 36 (not constant, try ratios instead)
How to recognize it
- Write out the differences between consecutive terms
- If differences are constant, you found the pattern
- If not constant, calculate differences of differences (second differences)
- Continue until you find a constant level
Common mistakes
- Stopping too early - sometimes you need second or third differences
- Using differences on geometric sequences (use ratios instead)
- Arithmetic errors in subtraction
Step-by-step walkthrough
Find the pattern in: 1, 4, 9, 16, 25
- Calculate first differences: 4-1=3, 9-4=5, 16-9=7, 25-16=9
- First differences not constant. Calculate second differences
- Second differences are constant (2) - quadratic pattern
- These are perfect squares: 1², 2², 3², 4², 5²
Answer: Perfect squares (n²)