Question

Find the general solution for a first-order homogeneous recurrence relation with constant coefficients. Is the restriction that the coefficients be constant necessary?

Short answer

The general solution is \( a_n = c^n \cdot a_0 \). Constant coefficients are necessary.

Step-by-step solution

Understanding the Problem

A first-order homogeneous recurrence relation with constant coefficients has the form \( a_n = c \cdot a_{n-1} \), where \( c \) is a constant coefficient. We need to find the general solution for this type of recurrence relation.

Recognize the Base Relation

The recurrence relation \( a_n = c \cdot a_{n-1} \) implies that each term is \( c \) times the previous term. This is a key insight into the pattern we need to describe.

Apply Iteration to Express Terms

To find the general solution, we can express several terms starting from an initial term, \( a_0 \). So, we have: - \( a_1 = c \cdot a_0 \) - \( a_2 = c \cdot a_1 = c^2 \cdot a_0 \) - \( a_3 = c \cdot a_2 = c^3 \cdot a_0 \) - and so on.

Derive the General Solution

From the pattern observed in Step 3, we can derive the general term \( a_n = c^n \cdot a_0 \). This expression describes every term \( a_n \) of the sequence in terms of the initial term \( a_0 \).

Discuss the Coefficient Requirement

In our solution, the constant coefficient \( c \) is crucial for the pattern \( a_n = c^n \cdot a_0 \) because it ensures the multiplicative buildup relation that defines each term in terms of its predecessor. If the coefficient were not constant, the pattern would not hold, and you'd need more complex forms to define each subsequent term.

Key concepts

Homogeneous Recurrence

A homogeneous recurrence relation is a sequence that defines each term using a specific pattern related to its preceding terms. In the context of a first-order recurrence relation like the one discussed here, each term is expressed as a multiple of its immediate predecessor. This can be equated to a sequence where the relation has eliminated any independent terms that do not involve the sequence's other terms.

The term 'homogeneous' signifies the absence of additional, unrelated terms or constants on the right side of the equation. This makes the structure simpler and allows the use of neat mathematical patterns to solve it. In this type of equation, if one term satisfies the equation, then any multiple of this term will fit as a solution. This is a fundamental property that aids in the solution of such sequences.

Constant Coefficients

In the context of recurrence relations, the idea of constant coefficients refers to a fixed numerical multiplier that links a term to its predecessor. For a first-order recurrence relation like \( a_n = c \cdot a_{n-1} \), the constant \( c \) plays a vital role.

These constant coefficients ensure that the relationship remains predictable and manageable throughout the sequence. They allow for an elegant, straightforward pattern where each term multiplies the previous term by the same value.

Without a constant coefficient, the predictability is lost, and solutions would require more complex methods which could involve varying calculations for different sequence elements. This fixed_multiplier simplification is a core feature that aids in deriving clear general solutions.

General Solution

The general solution of a homogeneous first-order recurrence relation with constant coefficients provides a succinct formula to compute any term in the sequence based on an initial condition. In the case covered in our example, the recurrence \( a_n = c^n \cdot a_0 \) gives a robust way to determine any term using the starting point, \( a_0 \), and \( c \), the constant coefficient.

Each sequence's term is neatly expressed in terms of \( n \), the position in the sequence. The expression \( c^n \cdot a_0 \) indicates that the term grows exponentially in \( n \) when \( c \) is greater than 1, reflecting rapid multiplications of the initial value.

• It shows how core arithmetic principles apply and simplify calculations.
• This form allows students to efficiently calculate large terms without needing intermediate steps.

Understanding the general solution empowers students to tackle various applications in computer science, economics, and natural sciences where recurrence relations model real-world problems.