Introduction
A Difference Series is a sequence where each term differs from the previous one by a (usually) consistent amount. Identifying the common difference or the missing terms is a frequent exam skill - it helps decode patterns quickly in number series, sequences and reasoning problems.
This pattern is important because it trains you to spot linear patterns, recover missing information, and check consistency in series questions.
Pattern: Difference Series (Finding Common Difference or Missing Terms)
Pattern
Key idea: If the difference between consecutive terms is constant, the sequence is an Arithmetic Progression (A.P.) with common difference d.
nth term formula (when a is first term and d is common difference):
Tₙ = a + (n - 1)·d
To find d: subtract any term from the next one: d = T₂ - T₁ = T₃ - T₂ = ….
To find a missing term: use the nth-term formula or fill sequentially using d.
Step-by-Step Example
Question
Find the missing term in the series: 7, __, 15, 19, 23.
Solution
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Step 1: Inspect consecutive differences
Compute differences where possible: 15 - ? and 19 - 15 = 4, 23 - 19 = 4. The last two differences are 4, so the common difference is likely d = 4.
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Step 2: Backfill missing term using d
Since 15 - d = missing term → 15 - 4 = 11. Check with first term: 7 → 7 + 4 = 11 (consistent).
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Step 3: Write the completed series
7, 11, 15, 19, 23.
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Final Answer:
Missing term = 11.
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Quick Check:
All consecutive differences: 11 - 7 = 4, 15 - 11 = 4, 19 - 15 = 4, 23 - 19 = 4 ✅
Quick Variations
1. Forward fill: Given a and d, produce terms using Tₙ = a + (n-1)d.
2. Backward fill: Given later terms, subtract d repeatedly to find earlier missing terms.
3. Multiple missing slots: Use index positions (e.g., if positions 2 and 4 missing) and the nth-term formula to set equations and solve.
4. Non-constant apparent differences: If differences vary, check second differences - constant second differences indicate a quadratic pattern (not A.P.).
Trick to Always Use
- Step 1 → Compute simple differences: subtract adjacent known terms to guess d.
- Step 2 → Confirm consistency: check at least two gaps to ensure d is constant.
- Step 3 → Use nth-term formula for non-adjacent misses: Tₙ = a + (n-1)d and solve for unknowns.
Summary
Summary
Key takeaways for Difference Series:
- Difference Series are usually A.P.s - identify the common difference d by subtracting consecutive terms.
- Use Tₙ = a + (n - 1)d to compute any term or solve for missing ones using positions.
- When differences are not constant, check second differences to detect non-linear patterns (quadratic sequences).
- Always perform a quick check by recomputing consecutive differences after filling missing values.
Hint: Subtract any term from the next to find d.
Common Mistakes: Doing differences in reverse order and getting negative values.