Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

The first two terms of the arithmetic sequence are given. Find the missing term. Use the table feature of a graphing utility to verify your results. $$a_{1}=-0.7, a_{2}=-13.8, a_{8}=\square$$

Step-by-Step Solution

Answer

The eighth term of the arithmetic sequence is -92.0

Step 1 - Calculate the common difference

The common difference, \(d\), in an arithmetic sequence can be found using the formula \(d = a_2 - a_1\), where \(a_2\) is the second term and \(a_1\) is the first term. Thus, \(d = -13.8 - (-0.7) = -13.1\).

Step 2 - Insert into the term formula

Now we can insert the numbers into the term formula. This will give the eighth term: \(a_{8} = a_{1} + (8-1) \cdot d = -0.7 + 7 \cdot(-13.1) = -92.0\).

Step 3 - Verify the result through table feature of a graphing utility

You can verify this result using a table feature of a graphing utility. Input the arithmetic sequence, with -0.7 as the first term and -13.1 as the common difference, and examine the eighth term.

Key Concepts

Common Difference

Understanding the common difference in an arithmetic sequence is essential for identifying the pattern that sequences follow. An arithmetic sequence is a list of numbers with a specific, constant step between each term. This step is known as the 'common difference' and is represented by the variable 'd'. For instance, in the sequence given in the exercise, we are provided with the first two terms, where the first term, \(a_1\), is -0.7 and the second term, \(a_2\), is -13.8. By utilizing the formula \(d = a_2 - a_1\), we find that the common difference is \(d = -13.8 - (-0.7) = -13.1\).

This common difference is the key to moving from one term to the next in the sequence. To find any term in the sequence, we add this common difference repeatedly. For example, to move from the first to the second term, we add \(d\) to \(a_1\). To move from the second to the third term, we add \(d\) to \(a_2\), and so on. Recognizing the role of the common difference helps students understand the linear nature of arithmetic sequences and how each term is related to its predecessors.

Graphing Utility

A graphing utility can serve as a powerful tool to visually confirm the behavior of arithmetic sequences and the calculations derived from them. The table feature in a graphing utility displays values that follow the sequence rules based on the established common difference. In our exercise, once we've calculated the common difference, \(d = -13.1\), we can use a graphing utility to plot the sequence.

The graphing utility needs two essential pieces of information: the first term and the common difference. From there, it generates subsequent terms. By reviewing the eighth term, \(a_8\), in the table, we can confirm that our manual calculation is correct. This verification step is a great way to double-check your work and strengthen your understanding of arithmetic sequences, ensuring that you've grasped the concept of how sequences progress based on their defined rules.

Term Formula

The term formula for an arithmetic sequence is a straightforward equation that allows you to calculate any term given its position within the sequence. Represented as \(a_n = a_1 + (n - 1) \times d\), where \(a_n\) is the n-th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference, we can plug in the appropriate values to determine any term.

For the exercise at hand, we are tasked with finding the eighth term, \(a_8\). Plugging the values into the term formula gives us \(a_{8} = -0.7 + (8 - 1) \times -13.1 = -92.0\). This formula reinforces the pattern-based nature of arithmetic sequences and provides a clear, methodical way to approach the calculation of terms in such sequences. By mastering the term formula, students can navigate through any arithmetic sequence and find specific terms with confidence.