A recursive formula defines a sequence of numbers in terms of its preceding terms. To determine the recursive formula for a specific sequence, you first need to identify the pattern or relationship between the elements of the sequence.
For example, let’s consider the Fibonacci sequence, which starts with 0 and 1. The recursive formula is defined as:
F(n) = F(n - 1) + F(n - 2)
with initial conditions:
F(0) = 0, F(1) = 1
This means each number in the Fibonacci sequence is the sum of the two preceding ones.
To construct a recursive formula for another sequence, you should do the following:
- Identify the starting point(s): Determine the first one or two terms of the sequence.
- Look for patterns: Investigate how the terms relate to each other. Are they being added, subtracted, multiplied, or divided?
- Define the recursive relation: Write the rule that describes how to get from one term to the next based on the identified patterns.
For instance, if we have a sequence where each term is 3 times the previous term, starting with 2, the recursive formula would be:
A(n) = 3 * A(n - 1)
with the initial condition:
A(1) = 2
In summary, the recursive formula captures the essence of how the sequence unfolds, linking each term back to its predecessors, which is crucial for both mathematical analysis and practical applications in programming and algorithm design.