Number Series Patterns

🎯 Learning Objective

Identify difference, ratio, square/cube and two-step patterns to find the next or missing term.

💡 Concept

• Step 1, always: write the differences between consecutive terms
• Equal differences (+d) → addition pattern; equal ratios (×r) → multiplication pattern
• Fast-growing series → think ×2, ×3 … or squares (1, 4, 9, 16, 25) and cubes (1, 8, 27, 64, 125)
• Two-step pattern: when first differences are unequal, take differences of the differences
• Alternating patterns (+2, +5, +2, +5) also appear — verify your rule on at least 3 terms

🧮 Key Formulas

Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

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Cubes: 1, 8, 27, 64, 125

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n(n+1) family: 2, 6, 12, 20, 30, 42

✏️ Easy Example

Q. Find the next term: 3, 7, 11, 15, ?

1. Differences: 7−3 = 4, 11−7 = 4, 15−11 = 4
2. Pattern is +4, so 15 + 4

Answer: 19

🇮🇳 Real-Life Example

Local trains leave at 6:10, 6:25, 6:40 — a +15 minute series. Spot the gap and you know the next train without the timetable.

📝 Exam-Level Example

Q. Find the missing term: 2, 6, 12, 20, 30, ?

1. Differences: 4, 6, 8, 10 — increasing by 2
2. Next difference = 12
3. 30 + 12 = 42

Answer: 42

📝 Exam-Level Example

Q. Find the next term: 5, 10, 30, 120, 600, ?

1. Ratios: ×2, ×3, ×4, ×5
2. Next ratio = ×6
3. 600 × 6 = 3600

Answer: 3600

🪄 Memory Trick

Differences equal? Done. Unequal? Take differences of differences. Growing too fast? Switch to ratios, squares or cubes.

⚠️ Common Mistakes

• ❌ Forcing a multiplication pattern when a simple +d works
• ❌ Stopping at unequal first differences instead of going one level deeper
• ❌ Missing alternating patterns like +2, +5, +2, +5

🏆 Exam Tips

• ✅ Memorise squares till 20 and cubes till 12 — series questions reuse them
• ✅ Verify your pattern on every given term, not just the first two

📌 Summary

• Always start with differences
• Unequal differences → two-step pattern
• Fast growth → ratio, square or cube
• Verify on 3+ terms before marking