Problem 8 - Essential Calculus Early Transcendentals

Question

\(5-8=\) Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. $$\\{5,8,11,14,17, \ldots\\}$$

Step-by-Step Solution

Answer

The general term is \(a_n = 3n + 2\).

Step 1: Identify the Pattern

Look at the given sequence: \(\{5,8,11,14,17, \ldots\}\). Notice that each term after the first is obtained by adding 3 to the previous term. This indicates that the sequence is an arithmetic sequence with a common difference \(d = 3\).

Step 2: Determine the First Term

Identify the first term of the sequence, which is \(a_1 = 5\). Since we have an arithmetic sequence where each term increases by 3, we start with this first term for our formula.

Step 3: Use the Formula for the nth Term of an Arithmetic Sequence

The general formula for the nth term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference. Substitute \(a_1 = 5\) and \(d = 3\) into this formula.

Step 4: Simplify the Formula

Substitute the known values into the equation: \(a_n = 5 + (n-1) \cdot 3\). Simplify this to find \(a_n = 5 + 3n - 3 = 3n + 2\). This is the formula for the nth term of the sequence.

Key Concepts

Common Difference

In arithmetic sequences, like the one you see in this exercise, the common difference is a key component. It's the value added to each term to get the next term in the sequence. For this particular sequence \(\{5,8,11,14,17, \ldots\}\), notice how each term increases by 3.
This consistent increase tells us that the common difference \(d\) is 3. Understanding the common difference helps us confirm that we’re dealing with an arithmetic sequence.

• Calculate: Subtract any term from the term that follows it. So here, \(8 - 5 = 3\), \(11 - 8 = 3\), and so on.
• Verify: Ensure this difference remains constant throughout the sequence. Consistency here is key.

General Term Formula

The general term formula, often written as \(a_n = a_1 + (n-1) \cdot d\), is fundamental in arithmetic sequences. This formula allows you to calculate any term in the sequence without listing them all out.
For the sequence given, where the first term \(a_1 = 5\) and the common difference \(d = 3\), we substitute these values into the formula:

• Substitute values: \(a_n = 5 + (n-1) \cdot 3\)
• Simplify: To make calculations easier, simplify to \(a_n = 5 + 3n - 3 = 3n + 2\).

This formula can now be used to find any term \(n\) in this sequence. For instance, if you want the 10th term, plug \(n = 10\) into the formula to get \(a_{10} = 3(10) + 2 = 32\).
The formula is a very efficient tool as it saves you time and effort.

Sequence Pattern Recognition

Recognizing patterns is crucial in dealing with sequences, especially when they appear in your math homework. For arithmetic sequences, like the one in this problem, the pattern is a linear increase or decrease by a constant.
Identifying and understanding this pattern early on makes it simpler to apply the general term formula. Here's how to recognize such patterns effectively:

• Observe Consecutive Terms: Look at how each term progresses from one to the next. For this sequence, each number increases by 3.
• Check Consistency: Ensure that this change is regularly maintained throughout the initial terms. This confirms the arithmetic nature of the sequence.
• Apply Knowledge: Use your understanding of the common difference and apply it in finding terms with the formula. This solidifies the pattern recognition process.

With practice, distinguishing these patterns becomes second nature, helping you solve arithmetic sequence problems with ease.