Recursive Sequence Definition
A mathematical process uses an initial value (or values) along with an explicit relationship to generate subsequent terms in a sequence. This iterative method defines each element after the first by applying a specific operation or function to the preceding element(s).
Defining a Sequence Recursively
- Initial Value(s): One or more starting terms are required to initiate the sequence. The number of initial terms depends on the relationship used.
- Recurrence Relation: This is an equation that expresses each subsequent term (an) as a function of one or more preceding terms (e.g., an-1, an-2).
- Iteration: The recurrence relation is applied repeatedly, using the previously calculated terms to find the next term in the progression.
Examples of Recursively Defined Sequences
Arithmetic Sequences
Defined by adding a constant difference (d) to the preceding term. For example: an = an-1 + d, with an initial value a1.
Geometric Sequences
Defined by multiplying the preceding term by a constant ratio (r). For example: an = r an-1, with an initial value a1.
Fibonacci Sequence
A classic example where each term is the sum of the two preceding terms. Defined as: an = an-1 + an-2, with initial values a1 = 0 and a2 = 1.
Applications
Recursive sequence definitions are found in various mathematical and computational contexts, including:
- Computer science (algorithm design, data structures)
- Finance (compound interest calculations)
- Physics (modeling natural phenomena)
- Mathematics (number theory, combinatorics)