Squares, Cubes, Primes & Fibonacci

🎯 Learning Objective

Recognise special number families — perfect squares, cubes, primes, Fibonacci-type and factorial-type series.

💡 Concept

• Perfect squares: 1, 4, 9, 16, 25, 36 … (n²). Often disguised as n²+1 or n²−1.
• Perfect cubes: 1, 8, 27, 64, 125 … (n³), sometimes as n³±1.
• Primes: 2, 3, 5, 7, 11, 13, 17 … (only two factors) — memorise them up to 50.
• Fibonacci-type: each term = sum of the previous TWO terms.
• Factorial-type: multiply by 2, then 3, then 4, then 5 … (1, 2, 6, 24, 120 …).

🧮 Key Formulas

Squares n²; cubes n³ (watch for ±1)

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Fibonacci: term = previous + one-before-previous

✏️ Easy Example

Q. Find the next term: 1, 4, 9, 16, ?

1. These are 1², 2², 3², 4²
2. Next is 5² = 25

Answer: 25

🇮🇳 Real-Life Example

The number of bricks to build square platforms of side 1, 2, 3 metres grows as 1, 4, 9 — perfect squares show up any time area is involved.

📝 Exam-Level Example

Q. Find the next term: 2, 5, 10, 17, 26, ?

1. Compare with squares: 1+1, 4+1, 9+1, 16+1, 25+1
2. So the rule is n² + 1
3. Next = 6² + 1 = 36 + 1 = 37

Answer: 37

📝 Exam-Level Example

Q. Find the next term: 2, 3, 5, 8, 13, ?

1. Check Fibonacci rule: 2+3 = 5, 3+5 = 8, 5+8 = 13
2. Each term = sum of previous two
3. Next = 8 + 13 = 21

Answer: 21

🪄 Memory Trick

Keep a mental table of squares up to 20² and cubes up to 12². When a term sits near a square/cube, test n²±1 or n³±1 immediately.

⚠️ Common Mistakes

• ❌ Missing the +1 or −1 shift over a clean square/cube
• ❌ Treating a Fibonacci series as an AP because early gaps look small
• ❌ Forgetting 2 is prime and the only even prime

🏆 Exam Tips

• ✅ Memorise squares 1–20 and cubes 1–12 cold
• ✅ For 'sum of previous two', confirm on at least three consecutive terms

📌 Summary

• Squares n², cubes n³ — watch for ±1
• Primes: 2,3,5,7,11,13,17,19,23,29 …
• Fibonacci → add the last two
• Factorial-type → ×2, ×3, ×4 …