Concept 15 of 19Advanced
Watch on YouTubeVideoRecursive patterns (each term from the previous)
A recursive rule defines each term using earlier ones:
- aₙ = 2·aₙ₋₁ + 1 starting a₁ = 1 → 1, 3, 7, 15, 31, 63, 127, …
- aₙ = aₙ₋₁ + aₙ₋₂ (Fibonacci) → 1, 1, 2, 3, 5, 8, 13, …
- aₙ = aₙ₋₁² starting 2 → 2, 4, 16, 256, 65536 (explosive!)
- aₙ = n·aₙ₋₁ starting 1 → 1, 2, 6, 24, 120 (factorials!)
Spot a recursive pattern when +/× don't work: ask "what operation turns one term into the next using n or previous terms?"
Example
5, 11, 23, 47, … pattern? Check 2·5+1=11 ✓, 2·11+1=23 ✓, 2·23+1=47 ✓. So aₙ = 2aₙ₋₁+1. Next: 2·47+1 = 95.
💡 Tip:Recursive rules are common in coding, compound interest, population growth.
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5 questions to check what you just read.
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- Q1.a₁ = 2, aₙ = aₙ₋₁ + 3. Find a₅.
- Q2.a₁ = 1, aₙ = 2·aₙ₋₁. Find a₅.
- Q3.Next in 3, 5, 9, 17, 33 (rule aₙ = 2·aₙ₋₁ − 1)?
- Q4.a₁ = a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂. Find a₆.
- Q5.Fibonacci F₁₀ in 1, 1, 2, 3, 5, 8, 13, 21, 34, 55?