Problem 42 - Beginning and Intermediate Algebra

Question

Find the number of terms in each arithmetic sequence. $$8,11,14,17, \ldots, 50$$

Step-by-Step Solution

Answer

The arithmetic sequence has \(n = 15\) terms.

Step 1: Identify the first term, common difference, and last term

From the arithmetic sequence, we can identify the following: - The first term \(a_1 = 8\) - The common difference \(d = 11 - 8 = 3\) - The last term \(a_n = 50\) Now we will plug these values into the arithmetic sequence formula to find the number of terms.

Step 2: Using the arithmetic sequence formula

The arithmetic sequence formula is as follows: \[a_n = a_1 + (n - 1)d\] In our situation, \(a_n = 50\), \(a_1 = 8\), and \(d = 3\). We want to find the value of \(n\), the number of terms in the sequence. Plugging the known values into the formula, we get: \[50 = 8 + (n - 1)3\]

Step 3: Solving for n

Now we have the equation and need to solve for \(n\). Follow these steps to find the value of \(n\): Step 1: Expand the equation: \[50 = 8 + 3n - 3\] Step 2: Simplify the equation: \[50 = 5 + 3n\] Step 3: Subtract 5 from both sides: \[45 = 3n\] Step 4: Divide by 3: \[n = 15\]

Step 4: Conclusion

The arithmetic sequence has \(n = 15\) terms.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a consistent number that is added to each term to get the next one. It is a foundational concept that determines the progression of the sequence. Let's explore this concept using the example from the exercise:

- The sequence is 8, 11, 14, 17, ..., 50.- The common difference (\(d\)) can be found by subtracting the first term from the second term: \(11 - 8 = 3\).

This means that each term increases by 3, allowing us to predict further terms in the sequence by continuous addition of this difference. Without a common difference, the sequence would not be arithmetic, as it relies on this uniform interval between terms to maintain its structure.

Number of Terms

Finding the number of terms in an arithmetic sequence involves determining how many terms fit between the first and the last term, given the common difference.

1. Identify the first term \(a_1\) (8 in this scenario), the common difference \(d\) (3), and the last term \(a_n\) (50).2. Use the formula and substitute the known values. For example, in this exercise, \(a_n = 50\), we solve for \(n\).

By utilizing the arithmetic sequence formula and solving for \(n\), we found there are 15 terms in this specific sequence. This understanding helps in further calculations and ensures clarity in how sequences develop.

Sequence Formula

The sequence formula for an arithmetic sequence is pivotal in solving problems related to terms and positions within the sequence. It provides a way to connect the physical terms we see with a mathematical relationship.

The formula is given by: \[a_n = a_1 + (n - 1)d\]

In this formula:

• \(a_n\) is the \(n\)th term.
• \(a_1\) is the first term.
• \(d\) is the common difference.

By plugging in known values, it allows us to find unknown components, which in this case was the number of terms, \(n\). This formula is an essential tool in understanding arithmetic progressions.

Problem Solving

To solve problems involving arithmetic sequences, it is crucial to follow a structured approach. This ensures that each step logically leads to the solution. Here is how you can tackle such problems:

- Start by identifying key components in the sequence: the first term, the last term, and the common difference.- Use the arithmetic sequence formula: \[a_n = a_1 + (n - 1)d\]- Substitute the known values into the formula.- Solve the equation for the unknown, such as finding \(n\) or another term.

In practice, like in the provided exercise, this step-by-step method helps students systematically uncover the desired information. It builds a robust understanding of how sequences operate and allows for confident application in various problems.