Problem 28 - Beginning and Intermediate Algebra

Question

For each arithmetic sequence, find \(a_{n}\) and then use \(a_{n}\) to find the indicated term. $$\frac{1}{3}, \frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}, \ldots ; a_{21}$$

Step-by-Step Solution

Answer

The common difference of the arithmetic sequence is \(\frac{1}{3}\), and using the formula \(a_n = a_1 + (n-1)d\), we find that the 21st term is \(a_{21} = 7\).

Step 1: Identify the common difference

The arithmetic sequence has a constant difference between each term. So, we need to find the difference between consecutive terms: Difference = \(\frac{2}{3} - \frac{1}{3} = \frac{1}{3}\) Note that the difference between every pair of consecutive terms is also \(\frac{1}{3}\).

Step 2: Use the arithmetic sequence formula

Now, we use the arithmetic sequence formula to find the nth term (\(a_n\)): \[a_n = a_1 + (n - 1)d\] Here, \(a_1\) is the first term of the sequence, \(n\) is the term number, and \(d\) is the common difference. In this case, \(a_1 = \frac{1}{3}\) and \(d = \frac{1}{3}\).

Step 3: Find \(a_{21}\)

We are given \(n = 21\) as the term to find. Plug the values into the formula: \[a_{21} = \frac{1}{3} + (21 - 1) \cdot \frac{1}{3}\]

Step 4: Simplify the expression

Now, we will simplify the expression to get our answer: \[a_{21} = \frac{1}{3} + 20 \cdot \frac{1}{3}\] \[a_{21} = \frac{1}{3} + \frac{20}{3}\] \[a_{21} = \frac{1+20}{3}\] \[a_{21} = \frac{21}{3}\] \[a_{21} = 7\]

Step 5: Conclusion

The 21st term in the given arithmetic sequence is \(a_{21} = 7\).

Key Concepts

Understanding the Common Difference

In arithmetic sequences, the common difference is a key concept. It is the constant value added or subtracted to each term in the sequence to get the next term. To identify it, you simply subtract a term from the one that follows it.
For example, if we have a sequence like \( \frac{1}{3}, \frac{2}{3}, 1, \frac{4}{3}, \ldots \), we find the common difference by subtracting \( \frac{1}{3} \) from \( \frac{2}{3} \).
Here's how it's done:

• Common difference = \( \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \)

The value \( \frac{1}{3} \) remains consistent between all consecutive terms of this sequence. This consistency is what defines the sequence as being arithmetic.

The nth Term Formula

The formula for finding the nth term of an arithmetic sequence is essential for solving many sequence-related problems. It allows you to find any term in the sequence without listing all the previous ones.
The general nth term formula is given by:

• \( a_n = a_1 + (n - 1)d \)

Let's break it down:

• \( a_n \) is the term we are looking for.
• \( a_1 \) is the first term in the sequence, which is \( \frac{1}{3} \) in our example.
• \( n \) represents the position of the term, for example, 21 for the 21st term.
• \( d \) is the common difference, which is \( \frac{1}{3} \).

Plug these values into the formula, and you can calculate any term in the sequence effortlessly.

Solving Arithmetic Sequence Problems

Arithmetic sequence problems can be tackled using a systematic approach. The key is understanding the formula and the components it uses.
To solve these problems:

• Identify the first term \( a_1 \) and the common difference \( d \).
• Apply the nth term formula \( a_n = a_1 + (n-1)d \) depending on the term you need.
• Solve the arithmetic expression that results from substituting the values into the formula.

With our example, finding the 21st term involves:- Using \( a_1 = \frac{1}{3} \), \( n = 21 \), \( d = \frac{1}{3} \)

• Calculating \( a_{21} = \frac{1}{3} + (21-1) \cdot \frac{1}{3} \).
• Simplifying to get \( a_{21} = 7 \).

This approach is reliable for any arithmetic sequence problems.

Sequence Simplification

Simplifying the arithmetic sequence expression is the final step in solving these types of problems. Once you've used the nth term formula, you often end up with a simple arithmetic expression.
For sequence simplification:

• Multiply the common difference by \( n-1 \) and add the result to the first term.
• Simplify any fractions or whole numbers, if necessary.
• Ensure the final answer is in its simplest form—whether it's a fraction or a whole number.

In the given example, the simplification was:

• Calculate \( \frac{1}{3} + 20 \cdot \frac{1}{3} \)
• Simplify to \( \frac{1+20}{3} \), then \( \frac{21}{3} \)
• Resulting in \( 7 \)

Effective simplification helps verify your solution's accuracy.