Problem 6 - Discrete Mathematics with Applications

Question

A_{n} denotes the \(n\) th term of a number sequence satisfying the given initial condition(s) and the recurrence relation. Compute the first four terms of the sequence. $$\begin{aligned} &a_{1}=1, a_{2}=2, a_{3}=3\\\ &a_{n}=a_{n-1}+a_{n-2}+a_{n-3}, n \geq 4 \end{aligned}$$

Step-by-Step Solution

Answer

The first four terms of the sequence are \(a_{1} = 1\), \(a_{2} = 2\), \(a_{3} = 3\), and \(a_{4} = 6\).

Step 1: We are given the first three terms of the sequence: \(a_{1} = 1\), \(a_{2} = 2\), and \(a_{3} = 3\). #Step 2: Analyze the recurrence relation #

The recurrence relation is given by: \(a_{n} = a_{n-1} + a_{n-2} + a_{n-3}\), for \(n \geq 4\). This means that to find the \(n\)th term of the sequence, we need to add the (\(n-1\))th, (\(n-2\))th, and (\(n-3\))th terms. #Step 3: Calculate the fourth term using the recurrence relation#

Step 2: To find the fourth term, \(a_{4}\), we will use the recurrence relation with \(n = 4\): \(a_{4} = a_{4-1} + a_{4-2} + a_{4-3} = a_{3} + a_{2} + a_{1}\) Now, substitute the initial conditions from Step 1: \(a_{4} = 3 + 2 + 1 = 6\) #Step 4: Present the first four terms of the sequence#

Based on the given initial conditions and the calculated fourth term, the first four terms of the sequence are: \(a_{1} = 1\), \(a_{2} = 2\), \(a_{3} = 3\), and \(a_{4} = 6\).

Key Concepts

Number Sequence

A number sequence is a set of numbers arranged in a specific order. Each number in a sequence is called a term. The order and arrangement of terms depend on the underlying rules or patterns governing the sequence. In this scenario, a number sequence begins with given terms and follows a specific rule or pattern described by a recurrence relation.
Number sequences are essential in mathematics as they allow us to predict subsequent terms based on a defined pattern. Sequences can be simple, like the sequence of natural numbers (1, 2, 3,...), or more complex, as in the sequence given in the exercise, which involves a specific recurrence relation and initial conditions.

• Understanding the pattern or rule of a sequence helps in calculating future terms.
• Sequences can differ in complexity based on their rules.
• This sequence type is defined by a recurrence relation, meaning each term depends on previous terms.

Initial Conditions

Initial conditions are the starting values in a number sequence, forming the basis from which other terms are calculated. In the context of the problem, the initial conditions are explicitly provided: \(a_{1}=1\), \(a_{2}=2\), and \(a_{3}=3\). These values serve as the foundation to apply the recurrence relation to determine subsequent terms.
Giving the initial conditions is crucial because they make it possible to compute further terms in the sequence accurately. Without these, you wouldn't have a starting point from which to apply any recurrence relation.

• Initial conditions determine the starting point of a sequence.
• They are necessary to apply any recurrence relation.
• In complex sequences, initial conditions act as the anchor to ensure accurate calculations.

Term Calculation

Calculating the terms of a number sequence requires using both the initial conditions and the recurrence relation. In this exercise, the recurrence relation is \(a_{n} = a_{n-1} + a_{n-2} + a_{n-3}\). To find a term like \(a_{4}\), we add the three previous terms:

\[a_{4} = a_{3} + a_{2} + a_{1} = 3 + 2 + 1 = 6\]
This process demonstrates how the recurrence relation allows for the calculation of any term in the sequence, provided you have the necessary preceding terms.

• Use the initial conditions as your starting point.
• Apply the recurrence relation to determine the new term.
• Repeat the process for calculating subsequent terms.

Mathematical Sequence Progression

Mathematical sequence progression refers to how a sequence evolves as each new term is added using rules set by the recurrence relation. Once the initial conditions are established, future terms can be calculated methodically. This progression reflects a sequence’s pattern over time.
In our example, the sequence progression follows adding the last three terms to predict the next one:

• Start with your initial conditions.
• Add the terms specified by the recurrence relation.
• Observe how the sequence grows or alters as new terms are introduced.

Understanding sequence progression helps in seeing the bigger picture of how the sequence develops, which is crucial for predicting long-term behavior or solving sequence-related problems.