Problem 6 - Discrete Mathematics with Applications

Question

In Exercises \(1-6, a_{n}\) 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. $$ \begin{array}{l}{a_{1}=1, a_{2}=2, a_{3}=3} \\\ {a_{n}=a_{n-1}+a_{n-2}+a_{n-3}, n \geq 4}\end{array} $$

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: Identify the known terms of the sequence

The exercise provided us with the initial conditions of the sequence, which are the first three terms. We have: \(a_1 = 1\), \(a_2 = 2\), and \(a_3 = 3\).

Step 2: Use the recurrence relation to find the fourth term

The recurrence relation given is \(a_n = a_{n-1} + a_{n-2} + a_{n-3}\), for \(n \geq 4\). To compute the fourth term, we plug in n=4: \[a_4 = a_{4-1} + a_{4-2} + a_{4-3}\] \[a_4 = a_3 + a_2 + a_1\]

Step 3: Calculate the value of the fourth term

Using the initial conditions, we substitute the values of the known terms into the equation: \[a_4 = a_3 + a_2 + a_1\] \[a_4 = 3 + 2 + 1\] \[a_4 = 6\]

Step 4: State the first four terms of the sequence

Now that we have calculated the fourth term, we can state the first four terms of the sequence: \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 3\), and \(a_4 = 6\).

Key Concepts

Understanding Number Sequences

Number sequences are a fundamental concept in mathematics, where each term in the sequence is defined according to a specific rule or formula. The beauty of number sequences lies in their predictability and structure. Once you understand the rule behind the sequence, you can predict future terms or even find terms much later in the sequence without having to compute every single one.

Consider the sequence of natural numbers: 1, 2, 3, 4, and so on. The rule here is simple: each term is one more than the previous term. But number sequences can be governed by more complex rules, such as arithmetic progressions, geometric progressions, or even more intricate patterns like the Fibonacci sequence, where each term is the sum of the two preceding ones. Understanding the underlying rule is key to mastering sequences.

The Role of Initial Conditions

Initial conditions in number sequences are like the coordinates on a treasure map; they pinpoint the exact starting point for our sequence journey. They are the first few terms given which, together with the recurrence relation, enable us to compute subsequent terms of the sequence.

In our exercise, the initial conditions are given as: \(a_1 = 1\), \(a_2 = 2\), and \(a_3 = 3\). These values are essential to begin the process of computing further terms. Think of initial conditions as 'seeds' from which the entire sequence 'grows'. Without these, it would be impossible to use the recurrence relation because you would have no terms to apply it to. Initial conditions provide the necessary groundwork to build the sequence.

Computing Sequence Terms

Once we are equipped with initial conditions and the recurrence relation, we can start computing terms in the sequence. This is like putting the key into the lock—the initial conditions—and turning it according to the mechanism—the recurrence relation—to unlock subsequent numbers in the sequence.

Following our exercise's recurrence relation \(a_n = a_{n-1} + a_{n-2} + a_{n-3}\), for \(n \geq 4\), we can see that each term from the fourth onwards is the sum of the three preceding terms. To calculate the fourth term, simply add up the first three: \(a_4 = 6\).

It’s like a chain reaction; once you find one term, you can use that to find the next, and so on. This process teaches us critical thinking and problem-solving, as we constantly use what we know to derive what we don't. With each step forward, the sequence builds upon itself, creating a beautiful progression of numbers.