Problem 12 - Algebra 2 and Trigonometry

Question

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots, n=7 $$

Step-by-Step Solution

Answer

The common difference is \( \frac{1}{2} \) and the 7th term is \( \frac{7}{2} \).

Step 1: Identify the Given Sequence and Terms

The given arithmetic sequence is \( \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots \). The task is to find the common difference and the 7th term (\( a_7 \)).

Step 2: Calculate the Common Difference

In an arithmetic sequence, the common difference \( d \) is found by subtracting any term from the following term. Here, calculate \( d = 1 - \frac{1}{2} = \frac{1}{2} \). Therefore, the common difference is \( \frac{1}{2} \).

Step 3: Write the General Formula for the n-th Term

The formula for the n-th 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. Here, \( a_1 = \frac{1}{2} \) and \( d = \frac{1}{2} \).

Step 4: Substitute Values to Find the 7th Term

Using the formula \( a_n = a_1 + (n-1) \cdot d \), substitute for \( a_1, d, \) and \( n \): \[ a_7 = \frac{1}{2} + (7-1) \cdot \frac{1}{2} \] Calculate this step-by-step: 1. \( 7-1 = 6 \)2. \( 6 \cdot \frac{1}{2} = 3 \)3. \( \frac{1}{2} + 3 = \frac{7}{2} \)

Step 5: Provide the Final Answer

The common difference is \( \frac{1}{2} \). The 7th term of the sequence is \( \frac{7}{2} \).

Key Concepts

Understanding the Common Difference

In any arithmetic sequence, the magic element is the common difference. It's what keeps the sequence progressing at a steady rate.
This common difference, often represented by the letter \(d\), is found by subtracting any term from the term that follows it.
For example, if we have the sequence \( \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots \), we can find the common difference by performing the calculation: \(d = 1 - \frac{1}{2} = \frac{1}{2}\).

• This means that each term in the sequence is \(\frac{1}{2}\) more than the previous one.
• This constant increase defines its arithmetic nature.

Understanding the common difference is crucial because it allows you to predict future terms and identify sequences that share similar properties.

The Power of the nth Term Formula

The nth term formula of an arithmetic sequence helps pinpoint any term’s position without listing all terms.
This swift and straightforward calculation is crucial when you need a specific term far down the sequence line.

The formula is: \(a_n = a_1 + (n-1) \cdot d\)
Where:

• \(a_n\) is the nth term you wish to calculate.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.

For our sequence, where \(a_1 = \frac{1}{2}\) and \(d = \frac{1}{2}\), to find the 7th term:
Substitute these into the formula:
\(a_7 = \frac{1}{2} + (7-1) \cdot \frac{1}{2}\)

This simplifies to: \(a_7 = \frac{1}{2} + 3 = \frac{7}{2}\). This step-by-step application showcases the efficiency of using the nth term formula.

Unveiling Sequence Analysis

Sequence analysis is about comprehending and interpreting the underlying patterns within a series of numbers.
For arithmetic sequences, identifying the common difference lays the groundwork for deeper analysis.
In our given sequence example:

• The consistent increase by \(\frac{1}{2}\) exemplifies a predictable linear pattern.
  Each successive term is a straightforward addition of the common difference to the previous term.
• This linear nature makes arithmetic sequences particularly easy to manipulate and understand.

With sequence analysis, you not only confirm the arithmetic nature but can also anticipate future patterns.
This is especially helpful in advanced math, coding algorithms, and even everyday logical problem solving.