Problem 17 - A Graphical Approach to College Algebra

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=8, a_{2}=6$$

Step-by-Step Solution

Answer

\(a_8 = -6\), \(a_n = 10 - 2n\).

Step 1: Identify the Common Difference

To find the common difference of an arithmetic sequence, we subtract the first term from the second term. Here, the first term \(a_1 = 8\) and the second term \(a_2 = 6\), so the common difference \(d\) is given by \(d = a_2 - a_1 = 6 - 8 = -2\).

Step 2: Formula for the nth Term

The formula for the nth term \(a_n\) of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \, d\). This formula helps in finding any term in the sequence when the first term and the common difference are known.

Step 3: Calculate \(a_8\)

To find the 8th term \(a_8\), substitute \(n = 8\) into the nth term formula: \(a_8 = a_1 + (8-1) \, d = 8 + 7(-2) = 8 - 14 = -6\). Therefore, the 8th term \(a_8 = -6\).

Step 4: Determine the General Formula for \(a_n\)

Substitute \(a_1 = 8\) and \(d = -2\) into the nth term formula: \(a_n = 8 + (n-1)(-2) = 8 - 2(n-1)\). Simplifying this gives \(a_n = 8 - 2n + 2 = 10 - 2n\). Thus, the general formula for the nth term is \(a_n = 10 - 2n\).

Key Concepts

Common Difference

In an arithmetic sequence, each term after the first is determined by adding a constant value called the "common difference." Understanding this concept is crucial as it sets the pattern for the sequence.
Let's consider the given sequence, where the first term is 8 and the second term is 6. To find the common difference, we subtract the first term from the second term:

• Common Difference (\(d) = a_2 - a_1 = 6 - 8 = -2\)

This negative common difference indicates that each subsequent term is 2 less than the previous term.
By knowing this, we can predict any term in the sequence by repeatedly subtracting 2 from the previous term. Understanding the common difference provides a foundation for calculating other terms in the sequence.

Formula for the nth Term

The formula for the nth term of an arithmetic sequence allows us to find any term without needing to list all preceding terms.
It is given by:

• \(a_n = a_1 + (n-1)d\)

Here, \(a_1\) is the first term, \(n\) is the term position, and \(d\) is the common difference. In our example:

• \(a_1 = 8\), \(d = -2\)

Substituting these values, the formula becomes:

• \(a_n = 8 + (n-1)(-2)\)

This formula is a powerful tool as it directly links the position of a term in the sequence to its value. For instance, if we need the 5th term, we simply plug \(n = 5\) into the formula. The ease with which we can find any term showcases the utility of this formula.

General Formula

The general formula is a simplified version of the nth term formula, specifically tailored for the given sequence.
By further simplifying:

• Start with \(a_n = 8 + (n-1)(-2)\)
• Distribute: \(a_n = 8 - 2(n-1)\)
• Expand and simplify: \(a_n = 8 - 2n + 2 = 10 - 2n\)

Hence, the general formula \(a_n = 10 - 2n\) expresses the nth term of this specific sequence in its simplest form.
This formula is incredibly useful as it allows quick computation of any term in the sequence by substituting different values of \(n\).
For example, to find \(a_{12}\), we calculate \(10 - 2(12)\), which simplifies to -14.
It streamlines finding terms compared to applying the basic nth term formula repeatedly.