Problem 69 - Algebra 2

Question

What is the first term in the arithmetic sequence? ____, \(8 \frac{1}{3}, 7,5 \frac{2}{3}, 4 \frac{1}{3}, \ldots\). A. 3 B. 9\(\frac{2}{3}\) C. 10\(\frac{1}{3}\) D. 11

Step-by-Step Solution

Answer

The first term is 9\(\frac{2}{3}\).

Step 1: Understand the problem

We are given an arithmetic sequence, which means each term increases or decreases by a constant difference. Our goal is to find the first term of the sequence given the terms: \(8 \frac{1}{3}\), \(7\), and \(5 \frac{2}{3}\).

Step 2: Identify the common difference

To find the common difference (d), subtract the second term from the first term of the sequence: \(d = 7 - 8 \frac{1}{3}\).

Key Concepts

common difference

An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This difference is called the 'common difference'. To find the common difference, subtract any term in the sequence from the one that follows it. For example, in the sequence given as \(8 \frac{1}{3}, 7, 5 \frac{2}{3}\), we can find the common difference by subtracting the second term from the first: \(d = 7 - 8 \frac{1}{3}\). This subtraction gives us the value of the common difference, which indicates how we transition from one term to the next. Knowing the common difference helps us determine the pattern of the sequence and enables us to find missing terms if needed.

• Key Point: Subtract any consecutive terms to find the common difference.

first term

The first term in an arithmetic sequence is simply the starting point from which the sequence begins. It's important because it sets the base for calculating other terms using the common difference. The challenge is to determine this initial value when it's not directly given, as in the exercise provided. To find the first term in the sequence like \(8 \frac{1}{3}, 7, 5 \frac{2}{3}\), once the common difference is known, we can manipulate the known terms to backtrack to this starting term. Often, calculating the difference backward from the known terms gives us a clear idea.

• Key Point: Knowing the first term helps construct the entire sequence.

sequence identification

Identifying arithmetic sequences involves recognizing the pattern of constant differences between terms. This pattern allows you to determine that you are indeed working with an arithmetic sequence. If no common difference can be established, then it might not be arithmetic. In our example, after identifying the sequence as arithmetic, we look for this common difference to confirm that it applies uniformly across all terms. This identification provides a structured view of how the sequence progresses, making it predictable and easier to work with.

• Key Point: Consistent difference means an arithmetic sequence.

step-by-step solution

Solving problems involving arithmetic sequences can often be simplified by tackling them in stages. Taking a step-by-step approach minimizes errors. Begin by understanding the problem itself; recognize the need for finding either the first term or another term within the sequence. Next, calculate the common difference by subtracting terms, as already established. Using this difference, you can then manipulate given terms to find the first or any missing term. Finally, verify your solution by checking if the established common difference applies consistently across the sequence.

• Key Point: A methodical approach leads to accurate results.