Recursive Sequence Generator
aₙ = p·aₙ₋₁ + q·aₙ₋₂
a₀ (first term)
a₁ (second term)
p (multiplier for aₙ₋₁)
q (multiplier for aₙ₋₂)
Terms to generate (≤30)
✓Sequence
1.00, 1.00, 2.00, 3.00, 5.00, 8.00, 13.00, 21.00, 34.00, 55.00
Last term
55.0000
Sum
143.0000
A recursive sequence defines each term using previous terms. The Fibonacci sequence is the most famous example (each term is the sum of the two before it). Many real-world processes follow recursive patterns.
- 1Define base cases: a₀, a₁
- 2Define recurrence: aₙ = f(aₙ₋₁, aₙ₋₂)
- 3For aₙ = p×aₙ₋₁ + q×aₙ₋₂
- 4Fibonacci is p=1, q=1
a₀=1, a₁=1, aₙ=aₙ₋₁+aₙ₋₂=1,1,2,3,5,8,13,21 (Fibonacci)
a₀=1, a₁=2, aₙ=2aₙ₋₁−aₙ₋₂=1,2,3,4,5,6 (arithmetic)
| Name | Rule | First 5 Terms |
|---|---|---|
| Fibonacci | aₙ=aₙ₋₁+aₙ₋₂ | 1,1,2,3,5 |
| Lucas | aₙ=aₙ₋₁+aₙ₋₂ | 2,1,3,4,7 |
| Geometric | aₙ=r×aₙ₋₁ | 1,r,r²,r³,r⁴ |
| Arithmetic | aₙ=aₙ₋₁+d | a,a+d,a+2d,... |
References
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