Mental Reasoning 2- Arithmetical Number Series: Questions and Step-by-Step Solutions for Competitive Exams SSC, BSSC, Bank, Railways

Mental Reasoning 2- Arithmetical Number Series: Questions and Step-by-Step Solutions for Competitive Exams SSC, BSSC, Bank, Railways

MENTAL REASONING BASED ON ARITHMETICAL NUMBER SERIES

1. What is an Arithmetical Number Series? (परिभाषा)

A Number Series is a sequence of numbers following a certain rule or pattern.
Your task is to identify that pattern and either:

• Find the next term,
• Find the missing term, or
• Find the wrong term.

2. Common Rules / Patterns (मुख्य नियम)

Let’s classify all possible and probable types seen in competitive exams.

Type 1: Constant Difference (Arithmetic Progression / A.P.)

Rule: Each term increases or decreases by a fixed number.

Examples:

1. 2, 4, 6, 8, 10, ? → +2 → 12
2. 100, 90, 80, 70, ? → –10 → 60

Shortcut:
→ Subtract consecutive numbers to check if the difference is constant.
Hindi: प्रत्येक संख्या में समान वृद्धि या कमी होती है।

Type 2: Constant Ratio (Geometric Progression / G.P.)

Rule: Each term is multiplied or divided by a fixed number.

Examples:

1. 2, 4, 8, 16, 32, ? → ×2 → 64
2. 81, 27, 9, 3, 1, ? → ÷3 → 1/3

Trick: Divide 2nd term by 1st; if ratio constant → G.P.

Type 3: Square / Cube Series (वर्ग / घन आधारित श्रेणी)

Rule: Terms are perfect squares or cubes or combinations.

Examples:

1. 1, 4, 9, 16, 25, ? → 36 (n² pattern)
2. 1, 8, 27, 64, ? → 125 (n³ pattern)
3. 2, 9, 28, 65, ? → 126 (n³ + 1 pattern)

Type 4: Difference of Differences (Second Difference Constant)

Rule: First differences are not equal but second differences are constant.

Examples:
2, 6, 12, 20, 30, ?
→ +4, +6, +8, +10 → +12 → 42

Hindi: लगातार पदों का अंतर बदलता है पर उसका अंतर (द्वितीय अंतर) समान होता है।

Type 5: Alternating Pattern (क्रमिक परिवर्तन)

Rule: Two or more patterns alternate.

Examples:

1. 2, 4, 8, 16, 18, 36, 38, ?
   → (×2, +2, ×2, +2) → Next: 76
2. 1, 3, 9, 11, 33, 35, ?
   → (×3, +2, ×3, +2) → 105

Type 6: Mixed Operations (+, –, ×, ÷)

Rule: Combination of different arithmetic operations.

Examples:

1. 5, 10, 20, 40, 80 → ×2
2. 3, 6, 12, 24, 48, 96 → ×2
3. 2, 5, 10, 17, 26, ?
   → +3, +5, +7, +9 → 38

Type 7: Prime Number Series (मूल संख्या आधारित श्रेणी)

Rule: Terms are prime numbers or follow prime increments.

Examples:

1. 2, 3, 5, 7, 11, 13, 17, ? → 19
2. 1, 2, 3, 5, 8, 13, 21, ? → Fibonacci → 34

Type 8: Fibonacci or Recursive Pattern (पूर्व पदों पर निर्भर)

Rule: Each term = sum or difference of previous one or two terms.

Examples:

1. 1, 1, 2, 3, 5, 8, 13, ? → 21
2. 2, 5, 7, 12, 19, ? → 2+5=7, 5+7=12, 7+12=19 → 31

Type 9: Square Difference Pattern (वर्गान्तर श्रेणी)

Rule: Differences follow a square pattern.

Example:
3, 4, 8, 15, 27, ?
→ +1, +4, +7, +12 → differences increase by +3, +3, +5 etc.

Or
1, 2, 6, 15, 31, ?
→ +1, +4, +9, +16 → difference = 1², 2², 3², 4² → 56

Type 10: Multiplication + Addition Combined

Rule: Multiply and add alternatively.

Examples:

1. 3, 7, 15, 31, 63, ?
   → ×2 +1 pattern → 127
2. 2, 6, 18, 54, ?
   → ×3 pattern → 162

Type 11: Odd-Even Position Pattern (विषम-सम स्थिति)

Rule: Odd and even positioned numbers follow different patterns.

Example:
3, 9, 5, 25, 7, 49, ?
→ Odd terms: 3, 5, 7 → +2
→ Even terms: 9, 25, 49 → squares of 3, 5, 7 → Next = 9², 49, 81 → 9,25,49,81 → Next term = 9 (odd term +2) → 9

Type 12: Number Reversal / Digit Pattern (अंकों की अदला-बदली)

Examples:

1. 12, 21, 13, 31, 14, 41, ? → 15, 51
2. 11, 22, 33, 44, ? → 55

Type 13: Arithmetic Based on Position or Multiplication of Digits

Examples:

1. 12, 24, 48, 96, ? → ×2 → 192
2. 3, 9, 27, 81, ? → ×3 → 243
3. 5, 15, 45, 135, ? → ×3 → 405

Type 14: Complex / Exam-style Logic (Multiple-Step Pattern)

Examples:

1. 2, 5, 10, 17, 26, 37, ?
   → Difference: +3, +5, +7, +9, +11 → Next = 50
2. 7, 14, 28, 56, 112, ?
   → ×2 pattern → 224
3. 10, 20, 40, 80, 160, 320 → ×2 pattern
4. 2, 6, 12, 20, 30, 42, ?
   → Differences = 4, 6, 8, 10, 12 → 56

Type 15: Wrong Number Series (त्रुटिपूर्ण संख्या श्रेणी)

Rule: Identify the term that breaks the pattern.

Example:
3, 6, 12, 24, 48, 97, 192
→ Correct pattern: ×2 → wrong = 97

Type 16: Decimal / Fractional Series (भिन्न श्रेणी)

Examples:

1. 1, 0.5, 0.25, 0.125, ? → ÷2 → 0.0625
2. 2, 4, 2, 4, 2, 4, ? → 2 (alternate)

Type 17: Difference in Multiples or Squares

Examples:
1, 4, 9, 16, 25 → n²
2, 3, 8, 15, 24, 35 → difference = 2,5,7,9,11 → odd increase
→ Next = 48

3. Strategy to Solve Arithmetical Number Series

4. Quick Practice Examples

5. Exam Tips for Arithmetical Number Series (परीक्षा में उपयोगी युक्तियाँ)

✅ Stepwise difference लिखें – इससे पैटर्न तुरंत दिखेगा।
✅ Alternate series का ध्यान रखें।
✅ Square/Cube, Prime, Fibonacci को याद रखें।
✅ जल्दी पहचानने के लिए mental math का अभ्यास करें।
✅ Wrong term वाले सवालों में pattern consistency चेक करें।
✅ Speed & accuracy बढ़ाने के लिए 100+ mixed series practice करें।