⭐ Sabse Important Points · एक नज़र में
- Constant Difference: If the gap between consecutive terms is the same, the series is arithmetic (e.g., 5, 9, 13, 17...).
- Alternating Series: Two interleaved sequences (e.g., odd positions follow one rule, even positions another).
- Constant Ratio: If each term is multiplied by a fixed number, it's geometric (e.g., 3, 6, 12, 24...).
- Double Difference: If first differences are not constant, check if their differences (second differences) are constant.
- Square/Cube Series: Terms close to perfect squares or cubes (e.g., 1, 4, 9, 16... or 1, 8, 27, 64...).
- Combined Operation: Pattern like ×2 + 1, ×2 + 2, ×2 + 3... where both multiplier and addend change.
Introduction · परिचय
Number Series is a high-frequency topic in SSC CHSL Reasoning. It tests your ability to identify the pattern governing a sequence of numbers and predict the missing or wrong term. Patterns range from simple arithmetic operations to complex combinations of differences, squares, cubes, and alternating rules.
Basic Patterns: Difference, Product, and Squares/Cubes
Most series follow a consistent mathematical rule. Start by checking differences between consecutive terms. If differences are constant, it's an arithmetic series. If ratios are constant, it's geometric. If differences grow rapidly, check squares or cubes. Also look for series where each term is derived from the previous by multiplying/dividing and adding/subtracting a constant.
- Check differences
- If constant → arithmetic
- If not, check ratios
- If constant → geometric
- If not, check squares/cubes
- If not, check mixed operations
- ⑂ condition ke hisaab se branch
📐 Formula / Rule Box · सूत्र याद रखो
nth term of arithmetic series
aₙ = a + (n-1)d
nth term of geometric series
aₙ = a × r^(n-1)
⭐ Key Points · याद रखने वाली बातें
- Constant Difference: If the gap between consecutive terms is the same, the series is arithmetic (e.g., 5, 9, 13, 17...).
- Constant Ratio: If each term is multiplied by a fixed number, it's geometric (e.g., 3, 6, 12, 24...).
- Square/Cube Series: Terms close to perfect squares or cubes (e.g., 1, 4, 9, 16... or 1, 8, 27, 64...).
- Mixed Operation: Pattern like ×2 + 1, ×2 + 3, etc., where multiplier and addend change systematically.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the next term: 2, 6, 12, 20, 30, ?
✅ Solution: 42
💡 Explanation: Differences: 4, 6, 8, 10 (increase by 2). Next difference = 12, so 30 + 12 = 42. Alternatively, terms are n(n+1): 1×2, 2×3, 3×4, 4×5, 5×6, 6×7=42.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Find the missing term: 3, 8, 15, 24, ?, 48
✅ Solution: 35
💡 Explanation: Differences: 5, 7, 9, 11, 13 (odd numbers). 24 + 11 = 35. Check: 35 + 13 = 48. Correct.
Advanced Patterns: Alternating, Double Difference, and Combined Operations
When simple patterns fail, series may alternate between two rules (e.g., +3, ×2, +3, ×2...), use double differences (differences of differences constant), or combine operations like ×2 + 1, ×3 + 2, etc. Also watch for series based on prime numbers, factorials, or digit-based operations.
- Check alternating positions
- If yes, solve two sub-series
- If no, check double differences
- If constant, extend
- If no, check combined operations
- If no, check primes/factorials
- ⑂ condition ke hisaab se branch
⭐ Key Points · याद रखने वाली बातें
- Alternating Series: Two interleaved sequences (e.g., odd positions follow one rule, even positions another).
- Double Difference: If first differences are not constant, check if their differences (second differences) are constant.
- Combined Operation: Pattern like ×2 + 1, ×2 + 2, ×2 + 3... where both multiplier and addend change.
- Prime/Factorial Series: Terms may be primes (2,3,5,7,11...) or factorials (1,2,6,24,120...).
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the next term: 5, 11, 23, 47, ?
✅ Solution: 95
💡 Explanation: Pattern: ×2 + 1, ×2 + 1, ×2 + 1... 5×2+1=11, 11×2+1=23, 23×2+1=47, 47×2+1=95.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Find the wrong term: 2, 5, 10, 17, 26, 37, 50
✅ Solution: All terms are correct; no wrong term (or if forced, pattern is n²+1: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17, 5²+1=26, 6²+1=37, 7²+1=50).
💡 Explanation: Check squares: 1,4,9,16,25,36,49 plus 1 gives 2,5,10,17,26,37,50. So series is correct.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Find the next term: 4, 9, 19, 39, 79, ?
👀 Answer dekho
✅ Answer: 159
💡 Explanation: Pattern: ×2 + 1, ×2 + 1, ×2 + 1... 4×2+1=9, 9×2+1=19, 19×2+1=39, 39×2+1=79, 79×2+1=159.
Q2Find the missing term: 1, 4, 9, 16, 25, ?, 49
👀 Answer dekho
✅ Answer: 36
💡 Explanation: Series of perfect squares: 1², 2², 3², 4², 5², 6²=36, 7²=49.
Q3Find the wrong term: 3, 7, 15, 31, 63, 127, 255
👀 Answer dekho
✅ Answer: No wrong term; all follow ×2 + 1.
💡 Explanation: 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1=63, 63×2+1=127, 127×2+1=255. Series is correct.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Assuming a single rule without testing alternatives: Always check differences, ratios, squares, and alternating patterns before concluding.
- Ignoring alternating sequences: If the series seems irregular, split into odd and even positions and check separately.
- Arithmetic errors in combined operations: Double-check calculations like ×2+1 or ×3-2 step by step.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Difference Method: First check constant difference; if not, check increasing/decreasing differences.
- Ratio Method: Check if each term is a fixed multiple of the previous.
- Square/Cube Recognition: Memorize squares up to 30 and cubes up to 15 for quick identification.
- Alternating Series: Look for two interleaved patterns when terms jump irregularly.
- Double Difference: If first differences form a pattern (e.g., arithmetic), extend to find next term.