0
0

Difference or Double Difference Series

Introduction

In a Difference or Double Difference Series, the difference between consecutive terms follows a pattern instead of the terms themselves forming a direct arithmetic or geometric progression. When the first-level differences are not constant, we check the second-level differences (also known as double differences) to uncover the pattern.

These series often appear in reasoning aptitude tests to evaluate a candidate’s ability to recognize underlying numerical patterns beyond simple addition or multiplication.

Pattern: Difference or Double Difference Series

Pattern

The main idea: if the difference between consecutive numbers keeps changing, find the difference of those differences (called the double difference).

In simple words, check how much each term increases or decreases. If that increase is not steady, look at how the increase itself changes - that’s your second difference.

Formulas (easy way):
1st Difference (D₁) = T₂ - T₁ → difference between two consecutive terms.
2nd Difference (D₂) = next D₁ - previous D₁ → difference between two consecutive first differences.
If all D₂ values are the same, the series follows a quadratic (square number) pattern.

Example pattern rule: Tₙ = a × n² + b × n + c (used when the second difference is constant).

Step-by-Step Example

Question

Find the next term in the series: 2, 5, 10, 17, 26, ?

Solution

  1. Step 1: Find first differences

    5 - 2 = 3, 10 - 5 = 5, 17 - 10 = 7, 26 - 17 = 9 → differences = 3, 5, 7, 9.

  2. Step 2: Check second differences

    5 - 3 = 2, 7 - 5 = 2, 9 - 7 = 2 → second difference is constant (+2).

  3. Step 3: Apply the rule

    Next first difference = 9 + 2 = 11 → Next term = 26 + 11 = 37.

  4. Final Answer:

    37

  5. Quick Check:

    1st diff: 3, 5, 7, 9, 11 → consistent +2 pattern ✅

Quick Variations

1. Constant first difference → Arithmetic Progression.

2. Constant second difference → Quadratic (Double Difference) Series.

3. Variable difference following arithmetic pattern → Higher order series.

4. Sometimes alternate patterns appear between even and odd positions.

Trick to Always Use

  • Step 1 → Compute the first difference of consecutive terms.
  • Step 2 → If not constant, compute the second (double) difference.
  • Step 3 → Add the next difference to the last term to find the next number.
  • Step 4 → For perfect quadratic sequences, check if n² fits the pattern.

Summary

Summary

  • When first difference is not constant, test second difference (D₂).
  • If D₂ is constant, series follows a quadratic relation.
  • Next term = last term + next first difference.
  • Check both odd and even term patterns in case of alternating differences.

Example to remember:
3, 6, 11, 18, 27 → next = 38 (+3, +5, +7, +9, +11)

Practice

(1/5)
1. Find the next term in the series: 4, 7, 10, 13, ?
easy
A. 16
B. 15
C. 18
D. 14

Solution

  1. Step 1: Find first differences

    7 - 4 = 3, 10 - 7 = 3, 13 - 10 = 3 → first differences = 3, 3, 3.

  2. Step 2: Observe pattern

    First differences are constant (+3) → it's an AP.

  3. Step 3: Apply the rule

    Next term = 13 + 3 = 16.

  4. Final Answer:

    16 → Option A

  5. Quick Check:

    Adding +3 each step: 4,7,10,13,16 ✅
Hint: If first differences are constant, it's an AP - just add the common difference.
Common Mistakes: Looking for a more complex rule when a simple AP exists.
2. Find the next term in the series: 1, 4, 9, 16, ?
easy
A. 25
B. 20
C. 24
D. 26

Solution

  1. Step 1: Try first differences

    4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7 → first differences = 3,5,7 (not constant).

  2. Step 2: Check second differences

    5 - 3 = 2, 7 - 5 = 2 → second differences constant = +2 → indicates a square pattern.

  3. Step 3: Apply the rule

    Recognize terms as perfect squares: 1², 2², 3², 4² → next = 5² = 25.

  4. Final Answer:

    25 → Option A

  5. Quick Check:

    Squares: 1,4,9,16,25 ✅
Hint: If second differences are constant (+2), check for perfect squares or quadratic form.
Common Mistakes: Assuming first differences must be constant before checking second differences.
3. Find the next term in the series: 20, 17, 13, 8, 2, ?
easy
A. -4
B. -6
C. -7
D. -5

Solution

  1. Step 1: Compute first differences

    17 - 20 = -3, 13 - 17 = -4, 8 - 13 = -5, 2 - 8 = -6 → first differences = -3, -4, -5, -6.

  2. Step 2: Check second differences

    (-4) - (-3) = -1, (-5) - (-4) = -1, (-6) - (-5) = -1 → second differences constant = -1.

  3. Step 3: Apply the rule

    Next first difference = -6 + (-1) = -7 → Next term = 2 + (-7) = -5.

  4. Final Answer:

    -5 → Option D

  5. Quick Check:

    First diffs: -3,-4,-5,-6,-7 → terms: 20,17,13,8,2,-5 ✅
Hint: Second differences can be negative - use the same rule: extend the second-difference to get the next first-difference.
Common Mistakes: Assuming second differences must be positive; negative constants are valid too.
4. Find the next term in the series: 2, 7, 15, 26, 40, ?
medium
A. 55
B. 57
C. 60
D. 58

Solution

  1. Step 1: Compute first differences

    7 - 2 = 5, 15 - 7 = 8, 26 - 15 = 11, 40 - 26 = 14 → first differences = 5, 8, 11, 14.

  2. Step 2: Check second differences

    8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3 → second differences constant = +3.

  3. Step 3: Apply the rule

    Next first difference = 14 + 3 = 17 → Next term = 40 + 17 = 57.

  4. Final Answer:

    57 → Option B

  5. Quick Check:

    First diffs: 5,8,11,14,17 (↑+3) → terms: 2,7,15,26,40,57 ✅
Hint: If second differences are constant but not +2, extend that second-difference to find the next first-difference.
Common Mistakes: Failing to check second differences when first differences are non-constant.
5. Find the next term in the series: 6, 10, 16, 24, 34, ?
medium
A. 44
B. 45
C. 46
D. 47

Solution

  1. Step 1: Compute first differences

    10 - 6 = 4, 16 - 10 = 6, 24 - 16 = 8, 34 - 24 = 10 → differences = 4, 6, 8, 10.

  2. Step 2: Check second differences

    6 - 4 = 2, 8 - 6 = 2, 10 - 8 = 2 → constant second difference = +2.

  3. Step 3: Apply the rule

    Next first difference = 10 + 2 = 12 → Next term = 34 + 12 = 46.

  4. Final Answer:

    46 → Option C

  5. Quick Check:

    First diffs: 4,6,8,10,12 → sequence: 6,10,16,24,34,46 ✅
Hint: If first differences grow by +2 each time, extend the pattern and add to last term.
Common Mistakes: Using +10 repeatedly instead of increasing the difference.

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes