Problem 20 - A Graphical Approach to College Algebra

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{15}=8, a_{17}=2$$

Step-by-Step Solution

Answer

The 8th term is 29, and the general term is \(a_n = 53 - 3n\).

Step 1: Understand the Problem

We are given an arithmetic sequence where the 15th term \(a_{15}\) is 8 and the 17th term \(a_{17}\) is 2. We need to find the 8th term \(a_8\) and a general formula for the \(n\)-th term \(a_n\).

Step 2: Determine the Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant. Since \(a_{17} = 2\) and \(a_{15} = 8\), we can calculate the common difference \(d\) by the formula: \[ d = \frac{a_{17} - a_{15}}{17 - 15} = \frac{2 - 8}{2} = -3 \] Thus, the common difference is \(-3\).

Step 3: Find the First Term

To find the first term \(a_1\), we use the formula for the \(n\)-th term: \(a_n = a_1 + (n-1) \times d\). For \(a_{15} = 8\), we set up the equation: \[ 8 = a_1 + 14(-3) \] Solving for \(a_1\), we get: \[ 8 = a_1 - 42 \]\[ a_1 = 50 \] So, the first term \(a_1\) is 50.

Step 4: Find the 8th Term

Use the \(n\)-th term formula to find \(a_8\): \[ a_{8} = a_1 + (8-1)d \] \[ a_{8} = 50 + 7(-3) \] \[ a_{8} = 50 - 21 = 29 \] Thus, the 8th term \(a_8\) is 29.

Step 5: Formulate the General Term \(a_n\)

Using the formula \(a_n = a_1 + (n-1)d\), substitute \(a_1 = 50\) and \(d = -3\): \[ a_n = 50 + (n-1)(-3) \] \[ a_n = 50 - 3n + 3 \] \[ a_n = 53 - 3n \] Therefore, the general formula for the \(n\)-th term \(a_n\) is \(53 - 3n\).

Key Concepts

Understanding Common Difference

In arithmetic sequences, the **common difference** is a key element that defines the uniform change from one term to the next. It's what makes the sequence "arithmetic." For any two consecutive terms \(a_{m}\) and \(a_{m+1}\), the common difference \(d\) can be found by the formula:

• \(d = a_{m+1} - a_{m}\)

This consistent "step" is crucial because it helps in forming the sequence and finding additional terms. In the given problem, the terms \(a_{15} = 8\) and \(a_{17} = 2\) allow us to compute the common difference, which turns out to be \(-3\). Knowing \(d = -3\), we understand each term decreases by 3 compared to the one before it. This information not only helps in finding previous and next terms but also assists in deriving other sequence properties.

Exploring the n-th Term Formula

The **n-th term formula** is a powerful equation that provides the value of any term in an arithmetic sequence. This formula is useful for constructing the sequence using the first term and common difference:

• \(a_n = a_1 + (n-1) \times d\)

In this equation:

• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term you want to find.
• \(d\) is the common difference.

For the example provided, substituting the known values gives us \(a_n = 50 + (n-1)(-3)\). Simplifying this, we get the general formula \(a_n = 53 - 3n\). This formula not only simplifies finding any particular term quickly but also reveals the linear nature of arithmetic sequences.

Calculating the First Term

The **first term calculation** is often the launching pad for understanding and applying concepts related to arithmetic sequences. Knowing the first term is crucial because it provides a starting point for applying the n-th term formula to find other terms. To find \(a_1\), we leverage a known term from the sequence along with the common difference.For instance, using \(a_{15} = 8\) and the common difference \(d = -3\), we can set up the formula \(a_{15} = a_1 + 14(-3)\). Solving this:

• \(8 = a_1 - 42\)
• \(a_1 = 50\)

Hence, the first term \(a_1\) is 50. Once the first term is known, it simplifies the process of finding any subsequent or preceding term using the n-th term formula. This makes navigation through the sequence much more straightforward.