How to Calculate Recursive Sequence
What is Recursive Sequence?
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.
Formula
General form: aₙ = f(aₙ₋₁, aₙ₋₂, ...) with initial conditions
- aₙ
- nth term of sequence
- aₙ₋₁, aₙ₋₂,...
- previous terms
- f
- recurrence relation function
Step-by-Step Guide
- 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
Worked Examples
Input
a₀=1, a₁=1, aₙ=aₙ₋₁+aₙ₋₂
Result
1,1,2,3,5,8,13,21 (Fibonacci)
Input
a₀=1, a₁=2, aₙ=2aₙ₋₁−aₙ₋₂
Result
1,2,3,4,5,6 (arithmetic)
Frequently Asked Questions
What is the difference between a recursive and explicit formula?
Recursive: defines aₙ using prior terms. Explicit: gives aₙ directly in terms of n.
Can every recursive sequence be expressed as explicit?
Not always easily. Some recursive sequences are difficult to express in closed form.
What are initial conditions in a recursive definition?
The first few terms (e.g., a₀ or a₁, a₂) that anchor the recursion so you can compute all later terms.
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