Problem 1 - 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. $$\begin{aligned} &a_{1}=1\\\ &a_{n}=a_{n-1}+3, n \geq 2 \end{aligned}$$

Step-by-Step Solution

Answer

The first four terms of the sequence are \(1, 4, 7, 10\).

Step 1: Write down the initial condition

Given the initial condition \(a_1 = 1\), the first term of the sequence is 1.

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

To find the second term of the sequence, we need to use the given recurrence relation. Since \(n = 2\), we have: \[a_2 = a_{2-1} + 3 = a_1 + 3\] Using the initial condition, we have: \[a_2 = 1 + 3 = 4\]

Step 3: Use the recurrence relation to find the third term

To find the third term of the sequence, we again use the given recurrence relation. This time, \(n = 3\), so we have: \[a_3 = a_{3-1} + 3 = a_2 + 3\] Using the second term that we just found, we have: \[a_3 = 4 + 3 = 7\]

Step 4: Use the recurrence relation to find the fourth term

Finally, to find the fourth term of the sequence, we use the given recurrence relation with \(n = 4\): \[a_4 = a_{4 - 1} + 3 = a_3 + 3\] Using the third term that we just found, we have: \[a_4 = 7 + 3 = 10\] So, the first four terms of the sequence are: \(1, 4, 7, 10\).

Key Concepts

Sequence

In mathematics, a sequence is a list of numbers arranged in a specific order. Each number in this list is referred to as a term. The fascinating aspect of sequences is that they can be finite or infinite. In other words, they might end after a few numbers, or they might go on forever. The sequence described in the exercise is one of the more common types, often called an arithmetic sequence.

• Each term after the first is generated by adding a constant value, known as the common difference, to the previous term.
• In this exercise, each term increases by 3.

A sequence is defined by its initial condition and a rule that determines the subsequent terms. Together, these define how the sequence evolves. Understanding sequences forms the backbone of various applications in not just mathematics, but also computer science and finance.

Initial Condition

The initial condition in a sequence provides the starting point of this list of numbers. It allows us to kick off the process of generating terms using the rules defined by the recurrence relation.

• For instance, the initial condition in the exercise specifies that the first term, \(a_1\), is equal to 1.
• It serves as a foundational reference from which all subsequent terms can be calculated.

Without an initial condition, the sequence cannot commence. You can think of it as the first domino that triggers the entire chain reaction. It holds significant importance as it ensures that all the terms are built consistently and correctly from a well-defined starting point.

Terms of a Sequence

The terms of a sequence refer to the individual numbers or elements that are generated by following the sequence's rule, starting from the initial condition. Each term is dependent on the term before it, following a specific pattern established by the recurrence relation. In our exercise example:

• The first term is 1, given by the initial condition.
• Each subsequent term adds 3 to the previous term.

Thus, the second term is 4, the third term is 7, and the fourth term is 10. To find any term in an arithmetic sequence like this, you can repeatedly apply the recurrence relation until you reach the desired term. The pattern remains consistent, reinforcing the predictable nature of a well-defined sequence.