How does the Recursive Sequence work?

Define base cases: a₀, a₁ Then: Define recurrence: aₙ = f(aₙ₋₁, aₙ₋₂) Then: For aₙ = p×aₙ₋₁ + q×aₙ₋₂ Then: Fibonacci is p=1, q=1.

Can you give an example of how to use the Recursive Sequence?

Example: Input a₀=1, a₁=1, aₙ=aₙ₋₁+aₙ₋₂ gives a result of 1,1,2,3,5,8,13,21 (Fibonacci).

Is the Recursive Sequence free to use?

Yes — completely free with no registration, download, or subscription required. All calculations happen instantly in your browser.

How accurate is the Recursive Sequence?

Our Recursive Sequence uses verified mathematical formulas and is accurate to multiple decimal places. Results are calculated in real-time using the same methods used by professionals.

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Recursive Sequence

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.

Symbol	Name	Unit	Description
aₙ	nth term of sequence	—	—
aₙ₋₁, aₙ₋₂,...	previous terms	—	—
f	recurrence relation function	—	—

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

›Recurrence relation — Wikipedia

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