Barisan Sebagai Solusi Relasi Rekursif

𝕸𝖆𝖘 𝕬𝖓𝖉𝖞 𝕯𝖎𝖌𝖎𝖙𝖆𝖑 𝕸𝖊𝖉𝖎𝖆

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Apr 12, 2025

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Sequences as Solutions to Recursive Relations: A Complete Recipe

Recursive relations are a cornerstone of mathematics and computer science, describing how a sequence of numbers is defined in terms of its preceding terms. Understanding how to solve these relations, often leading to explicit formulas for the sequence, is crucial for various applications. This post will provide a complete recipe for finding these solutions, focusing on techniques suitable for different types of recursive relations.

Understanding Recursive Relations

A recursive relation defines each term of a sequence based on previous terms. The most common form is:

a_n = f(a_n-1, a_n-2, ..., a_n-k)

where:

• a<sub>n</sub> is the nth term of the sequence.
• f is a function that describes the relationship between terms.
• k is the order of the relation (number of preceding terms involved).

For example, the Fibonacci sequence is defined by:

a_n = a_n-1 + a_n-2, with a₁ = 1 and a₂ = 1.

This is a second-order linear homogeneous recursive relation. Let's break down these terms:

• Order: Second-order because it involves two preceding terms.
• Linear: The terms a_n-1 and a_n-2 appear to the first power.
• Homogeneous: The function f does not contain a constant term independent of the a_i's.

Solving Linear Homogeneous Recursive Relations with Constant Coefficients

This is the most common type of recursive relation encountered. The general form is:

a_n = c₁a_n-1 + c₂a_n-2 + ... + c_ka_n-k

where c_i are constants.

The solution involves the characteristic equation:

r^k - c₁r^k-1 - c₂r^k-2 - ... - c_k = 0

Solving this equation (finding the roots r₁, r₂, ..., r_k) provides the key to finding the general solution.

Case 1: Distinct Real Roots

If all roots are distinct and real, the general solution is:

a_n = A₁r₁ⁿ + A₂r₂ⁿ + ... + A_kr_kⁿ

The coefficients A_i are determined using the initial conditions (the first few terms of the sequence).

Case 2: Repeated Real Roots

If a root r_i is repeated m times, the corresponding terms in the general solution become:

A_i1r_iⁿ + A_i2nr_iⁿ + ... + A_imn^m-1r_iⁿ

Case 3: Complex Roots

Complex roots always appear in conjugate pairs (a ± bi). Each pair contributes terms of the form:

rⁿ(Acos(nθ) + Bsin(nθ)), where r = √(a² + b²) and θ = arctan(b/a).

Solving Non-Homogeneous Recursive Relations

Non-homogeneous relations have a constant term or a term not involving a_i's. Solving these requires finding a particular solution and combining it with the complementary solution (obtained by treating the relation as homogeneous). Techniques like the method of undetermined coefficients can be useful here.

Example: Solving the Fibonacci Sequence

Let's solve the Fibonacci sequence (a_n = a_n-1 + a_n-2, a₁ = 1, a₂ = 1):

1. Characteristic equation: r² - r - 1 = 0
2. Roots: r₁ = (1 + √5)/2, r₂ = (1 - √5)/2 (the golden ratio and its conjugate)
3. General solution: a_n = A₁((1 + √5)/2)ⁿ + A₂((1 - √5)/2)ⁿ
4. Using initial conditions: We solve for A₁ and A₂ using a₁ = 1 and a₂ = 1. This gives us the explicit formula for the Fibonacci sequence.

Conclusion

Solving recursive relations provides a powerful way to understand and predict the behavior of sequences. This post outlined a step-by-step process, emphasizing various scenarios and techniques. Remember to practice with different examples to strengthen your understanding and build your skills. By mastering this skill, you’ll unlock many insights in mathematics, computer science, and other related fields.