Problem 27 - Precalculus: Mathematics for Calculus

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(2,4,6,8, \dots\)

Step-by-Step Solution

Answer

The nth term is \(a_n = 2n\).

Step 1: Identify the Pattern

Examine the sequence: 2, 4, 6, 8, ... and determine how each term relates to its position in the sequence. Notice that each term is 2 more than the previous term, suggesting a constant difference.

Step 2: Confirm the Pattern as an Arithmetic Sequence

Check if the pattern holds by finding the differences between consecutive terms: 4 - 2 = 2, 6 - 4 = 2, 8 - 6 = 2. Since the difference is constant, this is an arithmetic sequence.

Step 3: Derive the Formula for the General Term

Use the formula for the general term of an arithmetic sequence: \(a_n = a + (n-1) imes d\), where \(a\) is the first term and \(d\) is the common difference. Here, \(a = 2\) and \(d = 2\).

Step 4: Substitute Values into the Formula

Substitute \(a = 2\) and \(d = 2\) into the formula: \(a_n = 2 + (n-1) imes 2\). This simplifies to \(a_n = 2 + 2n - 2\), which further simplifies to \(a_n = 2n\).

Key Concepts

Common Difference

In mathematics, specifically in sequences, the *common difference* is a key component in determining whether a sequence is arithmetic. An arithmetic sequence occurs when the difference between consecutive terms is constant. For example, consider the sequence: 2, 4, 6, 8, ... This sequence is arithmetic because each subsequent term is obtained by adding 2 to the previous term. Here, 2 is the common difference. It's consistent across terms, allowing us to predict subsequent values in the sequence.

• The *common difference* is denoted as \(d\).
• If given the first term \(a\) and the common difference \(d\), you can generate the entire sequence.
• Recognizing this common difference helps distinguish the type of sequence you are dealing with, here it is arithmetic.

Identifying the common difference in a sequence provides a foundation for calculating any term in the sequence, without needing to list them all.

General Term Formula

To find any term in an arithmetic sequence, the *general term formula* is indispensable. The formula for the general term, or the \(n\)th term, of an arithmetic sequence is: \(a_n = a + (n-1) \times d\). Here, \(a\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
In our sequence example of 2, 4, 6, 8, ..., and using \(a = 2\) and \(d = 2\), the general term formula becomes \(a_n = 2 + (n-1) \times 2\). Simplifying this expression yields \(a_n = 2n\). This formula allows us to calculate any term in the sequence by simply plugging in the desired term number \(n\).

• Use \(a_n = a + (n-1) \times d\) to find any term.
• In the formula, replacing \(n\) with your desired term number gives its value.
• This approach saves time and effort since there's no need for listing all preceding terms.

Formulas such as this are tools that convert understanding from patterns into mathematical expressions that can be easily computed.

Sequence Pattern Recognition

Recognizing *patterns* in sequences is an important skill for identifying types of sequences and their properties. Pattern recognition encompasses observing how sequences progress and identifying if a predictable rule applies. For the arithmetic sequence 2, 4, 6, 8, ..., observe how each term increases by a constant value of 2. This consistent pattern tells us the sequence is arithmetic.
Understanding patterns allows us to

• Deducate the type of sequence (arithmetic, geometric, etc.).
• Predict subsequent terms without explicitly listing them.
• Formulate rules, like the general term formula, to find further terms efficiently.

Identifying and understanding patterns helps set a foundation for formulating mathematical expressions, which leads to solving complex problems with ease. Whether you're dealing with simple numeric sequences or more intricate ones, recognizing the underlying pattern is the first step in unraveling their behavior.