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Difference or Double Difference Series

Introduction

Difference or Double Difference Series में लगातार आने वाले terms के बीच का अंतर एक पैटर्न फॉलो करता है, बजाय इसके कि terms खुद किसी direct arithmetic या geometric progression का हिस्सा हों। जब first-level differences constant नहीं होते, तब हम second-level differences (जिन्हें double differences भी कहते हैं) को चेक करके असली pattern पहचानते हैं।

ऐसे series reasoning aptitude tests में अक्सर आते हैं क्योंकि ये candidate की ability को test करते हैं कि वह simple addition या multiplication से आगे बढ़कर hidden numerical patterns को पहचान सके।

Pattern: Difference or Double Difference Series

Pattern

मुख्य आइडिया: अगर लगातार numbers के बीच का difference बदलता रहता है, तो उनके differences का difference (जिसे double difference कहते हैं) निकालें।

आसान भाषा में-देखें कि हर term कितना बढ़ रहा है या घट रहा है। अगर यह बढ़ोतरी खुद भी तय नहीं है, तो देखें कि यह बढ़ोतरी कितनी बदल रही है - यही second difference है।

Formulas (easy way):
1st Difference (D₁) = T₂ - T₁ → दो लगातार terms का difference।
2nd Difference (D₂) = next D₁ - previous D₁ → दो लगातार first differences का difference।
अगर सारे D₂ values समान हों, तो series एक quadratic (square number) pattern फॉलो कर रही है।

Example pattern rule: Tₙ = a × n² + b × n + c (जब second difference constant होता है।)

Step-by-Step Example

Question

Series का अगला term बताइए: 2, 5, 10, 17, 26, ?

Solution

  1. Step 1: First differences निकालें

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

  2. Step 2: Second differences चेक करें

    5 - 3 = 2, 7 - 5 = 2, 9 - 7 = 2 → second difference constant (+2) है।

  3. Step 3: Rule लागू करें

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

  4. Final Answer:

    37

  5. Quick Check:

    1st diff: 3, 5, 7, 9, 11 → पूरा pattern +2 से बढ़ रहा है ✅

Quick Variations

1. Constant first difference → Arithmetic Progression।

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

3. Difference अगर खुद arithmetic rule फॉलो करे → Higher-order series।

4. कई बार odd और even positions पर अलग-अलग pattern चलते हैं।

Trick to Always Use

  • Step 1 → Consecutive terms का first difference निकालें।
  • Step 2 → Constant न होने पर second (double) difference निकालें।
  • Step 3 → Next difference को last term में जोड़कर अगला number पाएं।
  • Step 4 → Perfect quadratic sequences में n² pattern को match करें।

Summary

Summary

  • जब first difference constant न हो, तब second difference चेक करें।
  • अगर D₂ constant है, तो series quadratic होती है।
  • Next term → last term + next first difference।
  • Odd-even positions पर अलग patterns भी हो सकते हैं।

याद रखने वाला example:
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

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