Recursive Sequence: f(n+1) = -2f(n)
This article explores a recursive sequence defined by the formula f(n+1) = -2f(n). This formula describes a pattern where each term in the sequence is determined by the previous term, multiplied by -2.
Understanding the Formula
- f(n+1): Represents the (n+1)th term in the sequence.
- f(n): Represents the nth term in the sequence.
- -2: The constant factor by which each term is multiplied.
Exploring the Sequence
To understand the sequence, we need an initial value, often denoted as f(1). Let's assume f(1) = 3. Using the formula, we can generate the first few terms:
- f(2) = -2 * f(1) = -2 * 3 = -6
- f(3) = -2 * f(2) = -2 * (-6) = 12
- f(4) = -2 * f(3) = -2 * 12 = -24
- f(5) = -2 * f(4) = -2 * (-24) = 48
As you can see, the sequence alternates between positive and negative values, and the absolute value of each term is doubled.
Properties of the Sequence
- Geometric Sequence: This sequence is a geometric sequence because the ratio between consecutive terms is constant (-2).
- Alternating Signs: The signs of the terms alternate between positive and negative.
- Exponential Growth: The absolute value of the terms grows exponentially due to the constant multiplication by -2.
General Formula
We can find a general formula for the nth term of this sequence:
f(n) = f(1) * (-2)^(n-1)
This formula can be derived by repeatedly applying the recursive formula.
Applications
Recursive sequences like this have applications in various fields, including:
- Mathematics: Studying sequences and series.
- Computer Science: Implementing algorithms and analyzing data structures.
- Finance: Modeling compound interest and financial growth.
Conclusion
The recursive sequence defined by f(n+1) = -2f(n) demonstrates a simple yet powerful pattern that can be used to generate a sequence with alternating signs and exponential growth. Understanding this type of sequence provides valuable insights into the behavior of recursive formulas and their applications in different fields.
