Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the arithmetic series given two terms. \(a_{1}=17, \quad a_{7}=-31\)

Step-by-Step Solution

Answer

The first five terms are 17, 9, 1, -7, and -15.

Step 1: Understand the Given Terms

We are given two terms of an arithmetic series: the first term, \(a_1 = 17\), and the seventh term, \(a_7 = -31\). An arithmetic sequence has a constant difference (\(d\)) between consecutive terms.

Step 2: Use the General Formula

For an arithmetic sequence, the \(n\)-th term is given by \(a_n = a_1 + (n-1) \cdot d\). We can use this formula for the seventh term: \(a_7 = 17 + (7-1) \cdot d = -31\).

Step 3: Solve for the Common Difference

We can solve \(17 + 6d = -31\) to find \(d\). Rearranging gives us \(6d = -31 - 17\), so \(6d = -48\). Solving for \(d\), we divide by 6: \(d = \frac{-48}{6} = -8\).

Step 4: Find the First Five Terms

With \(a_1 = 17\) and \(d = -8\), we can find the first five terms: - \(a_1 = 17\)- \(a_2 = 17 - 8 = 9\)- \(a_3 = 9 - 8 = 1\)- \(a_4 = 1 - 8 = -7\)- \(a_5 = -7 - 8 = -15\).

Step 5: Verify Consistency

Check that the calculated terms are consistent with the original information. For example, the seventh term should be \(-31\), calculated as \(-31 = -15 + 2\times(-8)\). This matches the given \(a_7\).

Key Concepts

Common Difference

In the world of arithmetic sequences, the term 'common difference' plays a key role in understanding and constructing the sequence. Essentially, the common difference is the consistent amount we add or subtract from each term to reach the next term in the sequence. It is denoted by the symbol \(d\).

In our example, we started with the first term \(a_1 = 17\). We wanted to find the common difference that connects the first term \(a_1\) with the seventh term \(a_7 = -31\). By using the sequence formula \(a_n = a_1 + (n-1) \cdot d\), we plugged in all known values to solve for \(d\):

• The formula for the seventh term was set up as \(-31 = 17 + 6 \cdot d\).
• By rearranging, we found \(6d = -48\), leading to \(d = -8\).

This means each term in our sequence decreases by 8 to get the next term. Understanding the common difference helps address broader problems within arithmetic sequences.

Sequence Formula

The sequence formula is a tool for calculating any term in an arithmetic sequence, given the first term and the common difference. The general formula for an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). This lets us find any term in the sequence efficiently.

Let's break down its elements:

• \(a_n\): This represents the term you want to calculate.
• \(a_1\): This is the first term of the sequence. In our example, it is 17.
• \(n\): This is the term position you are interested in.
• \(d\): This symbolizes the common difference, which in our problem was calculated to be -8.

When you plug values into this formula, you seamlessly calculate any term within the sequence. For example, to find the fifth term \(a_5\), we substitute: \[a_5 = 17 + (5-1) \cdot (-8) = 17 - 32 = -15\]This powerful formula not only helps find specific terms but also confirms that a sequence is constructed correctly when checked against given term conditions.

Term Calculation

Calculating terms in an arithmetic sequence requires integrating both the common difference and the sequence formula. After identifying \(a_1\) and \(d\) from the problem, we can compute subsequent terms simply by applying the formula or by repetitive addition/subtraction of \(d\).

In our original exercise, with the parameters \(a_1 = 17\) and \(d = -8\), the process is broken down as follows:

• Start with \(a_1 = 17\).
• For \(a_2\), subtract 8 from the first term: \(9 = 17 - 8\).
• For \(a_3\), again subtract 8: \(1 = 9 - 8\).
• Continue this process to find \(a_4 = -7\) and \(a_5 = -15\).

This systematic approach ensures that you can quickly generate as many terms as needed, verify calculations, and understand the sequence's behavior. Each step reaffirms the linear pattern that defines arithmetic sequences, ultimately building a foundational understanding of how each part interacts within the sequence.