Problem 28 - Discrete Mathematics with Applications

Question

Using the recursive definition of \(c_{n},\) compute each. $$\mathcal{C}_{4}$$

Step-by-Step Solution

Answer

Given a linear recursive sequence defined as \(c_n = Ac_{n-1} + Bc_{n-2}\) and assuming initial values \(c_1 = 1\) and \(c_2 = 2\), we calculate the fourth term, \(c_4\), as follows: 1. Compute \(c_3 = Ac_2 + Bc_1 = A(2) + B(1)\). 2. Compute \(c_4 = Ac_3 + Bc_2 = A(c_3) + B(2)\). Substitute the given values for A, B, \(c_3\), and \(c_2\) in the formula to find the value of \(c_4\). Note that the actual solution depends on the provided recursive definition and initial term(s) in the exercise.

Step 1: Determine the Recursive Formula and the Initial Term(s)

First, let's find out the recursive definition of the sequence and determine the required initial terms to calculate the term \(c_4\). The information provided in the exercise may not be complete; therefore, we will assume a generic linear recursive sequence defined as: $$ c_n = Ac_{n-1} + Bc_{n-2} $$ where A, and B are constants. We will also need to know the first two terms of the sequence, \(c_1\) and \(c_2\). This information may not be given, so assuming some initial values:\(c_1 = 1\) and \(c_2 = 2\), for demonstration purposes.

Step 2: Compute the Third Term \(c_3\)

Using the recursive formula and the initial terms of the sequence, we can calculate the third term \(c_3\) as follows: $$ c_3 = Ac_2 + Bc_1 = A(2) + B(1) $$ We will need to plug in the values for A and B, as provided, to compute \(c_3\).

Step 3: Compute the Fourth Term \(c_4\)

Now that we have \(c_3\), we can compute \(c_4\) using the recursive formula in the same way: $$ c_4 = Ac_3 + Bc_2 = A(c_3) + B(2) $$ Again, we will need to plug in the values for A and B, as provided, to compute \(c_4\).

Step 4: Finalize the Solution

After substituting the values for A, B, \(c_3\), and \(c_2\) in the formula for \(c_4\), we obtain the desired value for \(c_4\). Please note that this illustration uses assumed values for A, B, \(c_1\), and \(c_2\), as these were not provided in the exercise. To solve this problem correctly, you must use the given recursive definition and initial term(s) provided in the exercise.

Key Concepts

Linear Recursion

Linear recursion is a method in which each term of a sequence is formulated based on certain fixed terms before it. This leads to the establishment of a sequence using previously defined terms, thereby following a consistent pattern. Linear recursive sequences are often defined by an equation, like the one below:\[ c_n = Ac_{n-1} + Bc_{n-2} \]In this formula:

• \(c_n\) is the current term in the sequence.
• \(c_{n-1}\) and \(c_{n-2}\) are the previous terms.
• \(A\) and \(B\) are constants that determine the contribution of each preceding term to the next term.

This recursive approach allows for a sequence to grow indefinitely without having to explicitly define every term initially; instead, only initial conditions and constants are necessary. Such sequences can model many natural processes and are widely used in mathematics and computer science. Breaking down sequences into simpler linear components often helps simplify complex problems and algorithms.

Initial Terms

Initial terms in a recursive sequence are the starting values required to begin the sequence. These are crucial because they serve as the base of the entire operation, and without them, further terms cannot be calculated.For the formula \( c_n = Ac_{n-1} + Bc_{n-2} \), knowing \( c_1 \) and \( c_2 \) allows you to compute all subsequent values of \( c_n \). These initial conditions form the foundation upon which recursion builds.

The Importance of Initial Terms

• Provide a starting point: Without initial terms, the recursive formula cannot operate.
• Determine the uniqueness: Different initial terms could lead to entirely different sequences, even if the recursive formula remains the same.
• Affect the sequence's behavior: The choice of initial terms can influence whether a sequence converges, diverges, or stabilizes.

Thus, when working with recursive sequences, it is vital to identify and understand the initial conditions accurately, because they impact the progression of the sequence.

Recursive Formula

A recursive formula defines a sequence using its preceding terms. This succinctly keeps track of the pattern or rule that every term follows, enabling computation of any term in the sequence from initial values.For example, the generic recursive formula:\[ c_n = Ac_{n-1} + Bc_{n-2} \]provides a clear method for generating each term:

• \(c_n\) is determined by combining \(Ac_{n-1}\) and \(Bc_{n-2}\).
• This formula allows for efficient computation without explicitly listing all terms.
• Binds the sequence to a defined progression dictated by initial values and coefficients \(A\) and \(B\).

Recursive formulas simplify calculations by eliminating repetitive calculations and reduce the need for external information beyond initial terms.

Benefits of Recursive Formulas

• Compact representation: The formula provides a concise way to describe the sequence.
• Scalable: Suitable for computing large sequences without redefining every term.
• Dynamic: Adapts to different situations by altering initial terms or coefficients.

Understanding and leveraging recursive formulas is crucial for effectively managing sequences and predicting future terms. They form the mathematical backbone for processes involving iteration and succession in various fields.