Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, find the specified term given two terms from an arithmetic sequence. \(a_{1}=33\) and \(a_{7}=-15\). Find \(a_{4}\).

Step-by-Step Solution

Answer

The fourth term, \(a_4\), is 9.

Step 1: Understand the Problem

We are given an arithmetic sequence with the first term as \(a_1 = 33\) and the seventh term as \(a_7 = -15\). We need to find the fourth term, \(a_4\). An arithmetic sequence is defined by a constant difference between consecutive terms.

Step 2: Determine the Common Difference

Use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1) imes d\), where \(d\) is the common difference. Substitute \(a_7 = -15\) to get:\[-15 = 33 + (7-1) imes d.\]Solve for \(d\):\[-15 = 33 + 6d.\]

Step 3: Solve for the Common Difference

Rearrange and solve the equation:\[-15 - 33 = 6d\]\[-48 = 6d\]\[d = \frac{-48}{6} = -8.\]Thus, the common difference \(d\) is \(-8\).

Step 4: Find the Fourth Term

Use the formula for the \(n\)-th term again, this time with \(n = 4\):\[a_4 = a_1 + (4-1) imes d = 33 + 3 imes (-8).\]Calculate \(a_4\):\[a_4 = 33 - 24 = 9.\]

Step 5: Verify the Sequence

To ensure correctness, verify the sequence:- \(a_1 = 33\)- \(a_2 = 33 - 8 = 25\)- \(a_3 = 25 - 8 = 17\)- \(a_4 = 17 - 8 = 9\)This sequence is consistent with a common difference of \(-8\).

Key Concepts

Understanding Common Difference in Arithmetic Sequences

In an arithmetic sequence, the common difference is a key element that makes the sequence arithmetic. This is the consistent difference between consecutive terms. Finding the common difference is crucial because it allows us to understand the pattern of the sequence. In our problem, the first term, \(a_1\), is 33, and the seventh term, \(a_7\), is -15.

By using these terms, we can calculate the common difference, \(d\). Insert the known values into the formula for the \(n\)-th term, \(a_n = a_1 + (n-1) \times d\). For \(a_7 = -15\), this becomes:

• \(-15 = 33 + 6 \times d\)
• Solving this, \(d = \frac{-48}{6} = -8\)
• Thus, the sequence decreases by 8 for each subsequent term.

Using the N-th Term Formula to Find Specific Terms

The \(n\)-th term formula, \(a_n = a_1 + (n-1) \times d\), is an incredibly helpful tool for arithmetic sequences. It enables us to find any term in the sequence given the first term and the common difference. In our exercise, after determining that the common difference, \(d\), is -8, we can easily find the fourth term, \(a_4\).

We substitute \(n = 4\), into the formula as follows:

• \(a_4 = 33 + 3 \times (-8)\)
• Simplifying this gives \(a_4 = 33 - 24\)
• Therefore, \(a_4 = 9\)

Using this formula, you can find any term in your sequence, as long as you know the first term and common difference.

Verifying the Arithmetic Sequence

Verification is a critical step to ensure the accuracy of your results. Once you have found the common difference and specific terms using the formula, it's wise to double-check your work by listing out the sequence. For our example, start with \(a_1 = 33\) and apply the common difference \(d = -8\) sequentially:

Consider each term:

• \(a_1 = 33\)
• \(a_2 = 33 - 8 = 25\)
• \(a_3 = 25 - 8 = 17\)
• \(a_4 = 17 - 8 = 9\)

This sequence consistently follows the calculated common difference, confirming that our calculations are correct. Verification helps to ensure the solution's reliability and strengthens your understanding.