Problem 11 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The closed-form solution for the given LHRRWCC is \(a_n = 2^n\).

Step 1: Define the generating function

Let's define the generating function \(A(x)\) as: \[A(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots\] Now our goal is to find \(A(x)\), so that we can later find the closed-form solution for the given LHRRWCC.

Step 2: Write the generating function equation

Let's multiply both sides of the given LHRRWCC by \(x^n\) and sum over all \(n \geq 1\): \[\sum_{n=1}^{\infty} a_nx^n = 2 \sum_{n=1}^{\infty} a_{n-1}x^n\] Now we can rewrite these sums in terms of the generating function, \(A(x)\): \[A(x) - a_0 = 2 \sum_{n=0}^{\infty} a_nx^{n+1}\] Since \(a_0 = 1\), we can write: \[A(x)-1 = 2 \sum_{n=0}^{\infty} a_nx^{n+1}\]

Step 3: Simplify the equation

Now, we can simplify the equation by factoring out the x on the right side: \[A(x) - 1 = 2x \sum_{n=0}^{\infty} a_nx^n\] This can be written in terms of \(A(x)\), since \(\sum_{n=0}^{\infty} a_nx^n = A(x)\): \[A(x) - 1 = 2xA(x)\] Now, we'll solve for \(A(x)\): \[A(x) = \frac{1}{1-2x}\]

Step 4: Find the closed-form solution

Finally, we will expand the generating function to find the closed-form solution of the given LHRRWCC. We can use the geometric series expansion for this: \[A(x) = \frac{1}{1-2x} = 1 + 2x + (2x)^2 + (2x)^3 + \cdots\] Comparing this with the definition of \(A(x)\), we get: \[a_n = 2^n\] Thus, the closed-form solution for the given LHRRWCC is \(a_n = 2^n\).

Key Concepts

Linear Homogeneous Recurrence Relation with Constant Coefficients

A linear homogeneous recurrence relation with constant coefficients (LHRRWCC) is a sequence defined such that each term is a linear combination of previous terms, and the coefficients that multiply these terms remain the same throughout the sequence. In simpler terms, it is like a recipe where each new term in the sequence is made by mixing a fixed portion of the earlier terms.

Consider the given recurrence relation:

• \(a_{n} = 2a_{n-1}\)
• with the initial condition \(a_0 = 1\)

This means each term is twice the previous term, and the process starts with 1.
These types of relations are called 'homogeneous' because the relation itself does not have independent terms; every part is made from the sequence itself.

Solving an LHRRWCC often involves techniques like generating functions which translate the problem into algebraic manipulations, making it easier to handle.

Closed-Form Solution

A closed-form solution refers to an explicit formula that directly calculates any term in the sequence without requiring previous terms.

In the exercise, we use generating functions to find a closed-form solution for the recurrence relation \(a_n = 2a_{n-1}\). The generating function creates a formal power series representing the entire sequence.

Following the simplification process in the steps, we derive:

• \(A(x) = \frac{1}{1-2x}\)

Using the geometric series expansion, this translates to:

• \(a_n = 2^n\)

This formula is a closed-form solution for the sequence, meaning you can find the value of \(a_n\) for any term \(n\) without needing to calculate the preceding terms, simplifying the computation significantly.

Geometric Series

A geometric series is a series of terms that each multiplies a fixed amount from the previous term. It has the general form of \(a, ar, ar^2, ar^3, ...\), where \(a\) is the first term and \(r\) is the common ratio between consecutive terms.

In generating functions, the geometric series plays a vital role because the sum of the series \(\frac{1}{1-r}\) equates to an infinite geometric series:

• 1 + r + r^2 + r^3 +...

In the provided solution, we apply this concept:

• \(A(x) = \frac{1}{1-2x}\)

Here, \(r = 2x\), which means the function expands to:

• 1 + 2x + (2x)^2 + (2x)^3 +...

Matching this with the sequence \(a_n\), the coefficients \((2x)^n\) show that the series grows exponentially, perfectly matching a geometric series. Thus, each term is simply two raised to the power of \(n\), providing insight into why the closed-form solution is \(a_n = 2^n\). The geometric series simplifies complex series relations into manageable terms to derive sequences and their closed forms.