Problem 17 - Discrete Mathematics with Applications

Question

Using generating functions, solve each LHRRWCC. $$a_{n}=5 a_{n-1}-6 a_{n-2}, a_{0}=4, a_{1}=7$$

Step-by-Step Solution

Answer

The closed-form solution of the given LHRRWCC is \(a_n = 10(2^n) - 6(3^n)\).

Step 1: Write down the generating functions

Let \(A(x)\) be the generating function for the sequence \(a_{n}\). That is, \[A(x) = a_{0} + a_{1}x + a_{2}x^2 + a_{3}x^3 + \cdots\] We will use the given recurrence relation and the initial conditions to transform \(A(x)\) into an algebraic equation and then solve for the closed form.

Step 2: Apply the recurrence relation to the generating function

Since \(a_n = 5a_{n-1} - 6a_{n-2}\), we can rewrite the generating function as follows: \[A(x) = a_0 + a_1x + \sum_{n=2}^{\infty} (5a_{n-1} - 6a_{n-2})x^n\] Now, let's split up this sum and find the generating functions for each part using the initial conditions. \[A(x) = 4 + 7x + 5x\sum_{n=2}^{\infty} a_{n-1}x^{n-1} - 6x^2\sum_{n=2}^{\infty} a_{n-2}x^{n-2}\]

Step 3: Rewrite the sums as generating functions

Now, rewrite the sums as generating functions: \[A(x) = 4 + 7x + 5x(A(x) - a_0 - a_1x) - 6x^2(A(x) - a_0)\] Plug in the initial conditions \(a_0 = 4\) and \(a_1 = 7\): \[A(x) = 4 + 7x + 5x(A(x) - 4 - 7x) - 6x^2(A(x) - 4)\]

Step 4: Solve for the generating function A(x)

Simplify and solve for \(A(x)\): \[A(x)(1 - 5x + 6x^2) = 4 + 7x - 20x - 35x^2 + 24x^2\] \[A(x) = \frac{4 + 7x - 20x - 35x^2 + 24x^2}{(1 - 5x + 6x^2)} = \frac{-16x^2 - 13x + 4}{(1 - 5x + 6x^2)}\]

Step 5: Find partial fraction decomposition of A(x)

We need to find the partial fraction decomposition of \(A(x)\): \[\frac{-16x^2 - 13x + 4}{(1 - 5x + 6x^2)} = \frac{A}{(1 - 2x)} + \frac{B}{(1 - 3x)}\] Solving for A and B, we get: \[A = 10\] \[B = -6\] So, \[A(x) = \frac{10}{(1 - 2x)} + \frac{-6}{(1 - 3x)}\]

Step 6: Find the closed-form solution of the sequence

Now, we will find the closed-form solution by applying the inverse of the generating function: \[a_n = 10(2^n) - 6(3^n)\] Thus, the closed-form solution of the given LHRRWCC is \(a_n = 10(2^n) - 6(3^n)\).

Key Concepts

Generating Functions

Generating functions are powerful tools used to solve recurrence relations. They act like a bridge between sequences and algebra. For a sequence \(a_n\), its generating function \(A(x)\) is expressed as \(A(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots\). This infinite series helps us treat the sequence as a whole object instead of individual terms.

With generating functions, you can manipulate sequences using algebraic operations. This ability is essential for finding solutions to recurrence problems like those involving sequences defined by linear relations.

Closed-Form Solution

A closed-form solution provides an explicit formula to compute any term in a sequence without needing the previous terms. In our recurrence relation, we used generating functions to find:

\[a_n = 10(2^n) - 6(3^n)\]

This formula allows us to calculate \(a_n\) directly. It frees us from the repetitive calculations required if we only had the recurrence relation.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify complex rational expressions. When we have a rational function like \(\frac{-16x^2 - 13x + 4}{(1 - 5x + 6x^2)}\), decomposition breaks it into simpler fractions.

In our solution, we divided it into:

• \(\frac{10}{(1 - 2x)}\)
• \(\frac{-6}{(1 - 3x)}\)

This separation makes it easier to work with the generating function, allowing us to apply the inverse and find the closed-form solution.

Linear Homogeneous Recurrence Relation with Constant Coefficients

A linear homogeneous recurrence relation with constant coefficients (LHRRWCC) is a specific type of recurrence relation. It has constant coefficients and the relation only involves terms of the sequence itself and not any added functions of the index \(n\).

Our given relation is:
\(a_n = 5a_{n-1} - 6a_{n-2}\)
Such relations often arise in mathematical modeling and can be efficiently solved using generating functions, as demonstrated. The efficient resolution paves the way for deeper insights into the behavior of sequences.