Problem 50 - Precalculus with Limits

Question

In Exercises 47 - 50, the first two terms of the arithmetic sequence are given. Find the missing term. \( a_1 = -0.7, a_2 = -13.8, a_8 = \)

Step-by-Step Solution

Answer

The missing term \(a_8\) in the arithmetic sequence is -92.4.

Step 1: Calculate the common difference

The common difference \(d\) of an arithmetic sequence can be found by subtracting the first term \(a_1\) from the second term \(a_2\). Therefore, \(d = a_2 - a_1 = -13.8 - (-0.7) = -13.8 + 0.7 = -13.1 .\)

Step 2: Use the formula to find the missing term

Now that we have the common difference, we can substitute it into the arithmetic sequence formula along with the provided first term and the term position we are looking for to find the value of \(a_8\). The formula is \(a_8 = a_1 + (8-1) \cdot d = -0.7 + 7 \cdot -13.1 = -0.7 - 91.7 = -92.4 .\)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is pivotal. It is the consistent interval between consecutive terms in a sequence. The common difference, denoted as \( d \), helps shape the entire progression of the sequence. To determine \( d \), simply subtract one term from the next. In our exercise, we calculated it as follows:

• First term \( a_1 = -0.7 \)
• Second term \( a_2 = -13.8 \)
• Common difference \( d = a_2 - a_1 = -13.8 - (-0.7) = -13.1 \)

This consistent change, \( -13.1 \), shows how much each term decreases as you move from one to the next in this particular sequence.

Arithmetic Sequence Formula

The arithmetic sequence formula is the mathematical tool used to find any term in a sequence. It is expressed as:\[a_n = a_1 + (n-1) \cdot d\]It helps us figure out the value of any term \( a_n \) by using:

• \( a_1 \): The first term in the sequence
• \( n \): The position of the term you need to find
• \( d \): The common difference

For instance, in the exercise, we needed to find \( a_8 \). Using the formula, we determined:\[a_8 = -0.7 + (8-1) \cdot (-13.1) = -92.4\]Understanding this formula allows you to harness the structure of arithmetic sequences efficiently.

Terms of a Sequence

In any arithmetic sequence, each number is known as a term. These terms follow a specific order, and knowing how they progress helps in understanding the nature of the sequence. Terms are usually represented by \( a_n \), where \( n \) is their position within the sequence. Each term is directly linked to its predecessor and successor by the common difference.For example, in our sequence:

• \( a_1 = -0.7 \)
• \( a_2 = -13.8 \)
• \( a_3 \)= determined by adding \( d \) to \( a_2 \)

The eighth term was calculated using the arithmetic sequence formula, showing how these terms systematically derive from each other. Terms are the building blocks, showing the path across the sequence.