Problem 16 - Calculus: Early Transcendentals

Question

Find a formula for the general term \( a_n \) of the sequence, assuming that the pattern of the first few terms continues. \( \left\\{\begin{array} 5, 8, 11, 14, 17, . . . . .\end{array}\right\\} \)

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern

Look for a pattern in the sequence: 5, 8, 11, 14, 17. Notice that each term increases by 3 from the previous term. This indicates that the sequence is arithmetic.

Step 2: Determine the First Term

The first term in the sequence is given as 5. Let this be denoted as \( a_1 = 5 \).

Step 3: Determine the Common Difference

The common difference \( d \) can be determined by subtracting any term from the term that follows it. For example, \( 8 - 5 = 3 \). Thus, the common difference is 3.

Step 4: Use the Formula for the General Term of an Arithmetic Sequence

The formula for the \( n^{th} \) term of an arithmetic sequence is given by \( a_n = a_1 + (n - 1) \times d \).

Step 5: Substitute Known Values into the Formula

Substitute \( a_1 = 5 \) and \( d = 3 \) into the formula: \( a_n = 5 + (n - 1) \times 3 \).

Step 6: Simplify the Expression

Simplify the expression: \[ a_n = 5 + 3n - 3 \]. Combine like terms to get \( a_n = 3n + 2 \).

Key Concepts

Common Difference

In arithmetic sequences, the "common difference" is a pivotal concept. It represents the consistent amount by which each term in the sequence increases or decreases from the previous term. To find this common difference, simply subtract one term from the term that follows it.
In our example sequence: 5, 8, 11, 14, 17, the common difference is found as follows:

• Subtract the first term from the second: \( 8 - 5 = 3 \)
• Subtract the second term from the third: \( 11 - 8 = 3 \)
• This pattern of "difference" remains constant for all consecutive terms, confirming it is indeed an arithmetic sequence.

Understanding the common difference helps you not only identify the sequence type but also lays the groundwork for calculating the general term.

General Term Formula

The general term formula of an arithmetic sequence allows you to quickly determine the value of any term without listing all predecessors. This formula expresses the relationship of each term to its position in the sequence. It's written as \( a_n = a_1 + (n - 1) \times d \):

• \( a_n \) represents the term we want to find.
• \( a_1 \) is the first term in the sequence.
• \( n \) is the position number of the term.
• \( d \) is the common difference.

By understanding and using this formula, you can find any term rapidly. Let's apply it to our sequence. With \( a_1 = 5 \) and \( d = 3 \), substitute these into the formula:
\[ a_n = 5 + (n - 1) \times 3 \]
It is often simplified to \( a_n = 3n + 2 \), showing that the term increases by a factor of 3 with each step forward in the sequence position.

Sequence Pattern Identification

Identifying the pattern of a sequence is crucial. It serves as the first step in solving any problems related to sequences. For arithmetic sequences, this usually means spotting the regular intervals at which terms increase or decrease.
Here’s how you identify a sequence pattern in three simple steps:

• Examine the sequence closely. Compare consecutive terms.
• Calculate the difference between the terms: \( 8 - 5 = 3, \) \( 11 - 8 = 3 \), and so on. Constant results mean you have an arithmetic sequence.
• Confirm by repeatedly checking that the difference remains the same for all steps in the given sequence.

Once you confirm a consistent difference across terms, you've effectively identified the sequence's pattern. This pattern, known as the common difference, is your key to unlocking the general term and understanding the sequence’s behavior.