Arithmetic & Geometric Patterns

🎯 Learning Objective

Recognise constant-difference (AP), constant-ratio (GP) and combined ×n±k patterns and find the next term.

💡 Concept

• Arithmetic Progression (AP): each term differs from the previous by a CONSTANT difference (add/subtract the same number).
• Geometric Progression (GP): each term is the previous one MULTIPLIED by a constant ratio.
• First check the difference between terms; if it is constant, it is an AP.
• If differences grow fast (doubling, tripling), suspect a GP or a ×n±k rule.
• Common exam pattern: each term = previous × 2 + 1 (or similar mixed operation).

🧮 Key Formulas

AP: next = last + d (common difference d)

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GP: next = last × r (common ratio r)

✏️ Easy Example

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

1. Difference: 7−3 = 4, 11−7 = 4, 15−11 = 4 (constant)
2. It is an AP with d = 4
3. Next = 15 + 4 = 19

Answer: 19

🇮🇳 Real-Life Example

Railway platform numbering, milestone markers every 4 km on a highway, monthly SIP of a fixed ₹500 — all march up in arithmetic steps, term after term.

📝 Exam-Level Example

Q. Find the next term: 5, 11, 23, 47, ?

1. Check ratio/rule: 5×2 + 1 = 11, 11×2 + 1 = 23, 23×2 + 1 = 47
2. The rule is ×2 + 1
3. Next = 47×2 + 1 = 95

Answer: 95

📝 Exam-Level Example

Q. Find the next term: 7, 10, 16, 28, ?

1. Differences: 3, 6, 12 — each difference is doubling
2. Next difference = 12 × 2 = 24
3. Next term = 28 + 24 = 52

Answer: 52

🪄 Memory Trick

Always write the differences first. Constant → AP. Doubling/tripling differences → GP or a ×n±k rule. This one step classifies most series.

⚠️ Common Mistakes

• ❌ Assuming AP without checking every gap (one term can hide a different rule)
• ❌ Confusing a ×2+1 rule with a pure GP (×2)
• ❌ Looking only at the last two terms instead of the whole pattern

🏆 Exam Tips

• ✅ Test the simplest rule first: +d, then ×r, then ×n±k
• ✅ If a single difference breaks the pattern, look at second-level differences

📌 Summary

• AP → constant difference; GP → constant ratio
• Write differences before guessing
• Doubling gaps hint ×2 or ×2+1 rules
• Verify the rule on ALL given terms