Problem 28 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) for each arithmetic sequence. $$a_{6}=-8, a_{7}=-18$$

Step-by-Step Solution

Answer

\(a_1 = 42\)

Step 1: Understand Arithmetic Sequence Formula

In an arithmetic sequence, each term after the first is the sum of the previous term and a constant known as the common difference, denoted as \(d\). The general formula for the \(n\)-th term of an arithmetic sequence is: \( a_n = a_1 + (n - 1)d \).

Step 2: Set Up Equations for Given Terms

Using the formula \( a_n = a_1 + (n - 1)d \), we set up equations for \( a_6 \) and \( a_7 \). Thus, \( a_6 = a_1 + 5d = -8 \) and \( a_7 = a_1 + 6d = -18 \).

Step 3: Subtract Equations to Find the Common Difference

Subtract the equation for \( a_6 \) from \( a_7 \):\[ (a_1 + 6d) - (a_1 + 5d) = -18 - (-8) \]This simplifies to:\[ d = -10 \]

Step 4: Solve for \( a_1 \) Using the Common Difference

Substitute \( d = -10 \) back into one of the equations to find \( a_1 \). Using \( a_6 = a_1 + 5d = -8 \):\[ a_1 + 5(-10) = -8 \]\[ a_1 - 50 = -8 \]\[ a_1 = 42 \]

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a fundamental concept. It's the amount by which each term in the sequence increases (or decreases) to reach the next term. You can think of it as the step size between numbers in the sequence.

When you know the common difference, you can easily predict future terms in the sequence. For example, in the problem given, we found the common difference, \(d\), by looking at terms \(a_6\) and \(a_7\):

• \(a_6 = -8\)
• \(a_7 = -18\)

The subtraction of these terms gives the common difference. This serves as the stepping stone to solve the entire sequence problem. It's vital to understand how this difference helps to construct the sequence.

Arithmetic Sequence Formula

The arithmetic sequence formula is your best friend when it comes to exploring arithmetic progressions. This formula helps in finding any term in the sequence as long as you know the first term and the common difference. The formula is:

• \(a_n = a_1 + (n - 1)d\)

Here, \(a_n\) is the \(n\)-th term you want to find, \(a_1\) is the first term, \(n\) is the position of the term in the sequence, and \(d\) is the common difference.

For example, to find \(a_6\) based on the known values given in the problem, we set up the equation \(a_6 = a_1 + 5d\). By using this formula, you can backtrack or forecast terms in a sequence seamlessly.

Problem Solving Steps

Solving problems related to arithmetic sequences involves specific steps which make the process manageable and straightforward. Let's unravel the step-by-step method using the example task:

• Identify the knowns: Pin down given terms such as \(a_6 = -8\), and \(a_7 = -18\).
• Use the Arithmetic Sequence Formula: Write down the expressions for the given terms, such as \(a_6 = a_1 + 5d\).
• Calculate the Common Difference: Subtract the equations to find \(d\), in this case from \(a_7 - a_6\), yielding \(d = -10\).
• Solve for the first term: Substitute the common difference into one of the equations to solve for \(a_1\), arriving at \(a_1 = 42\).

Emphasizing such steps ensures clarity and consistency in solving arithmetic sequence problems. Following this guide will help you solve these tasks with more confidence and accuracy.