Problem 18 - A Graphical Approach to College Algebra

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=6, a_{2}=3$$

Step-by-Step Solution

Answer

\(a_{8} = -15\); \(a_{n} = 9 - 3n\).

Step 1: Identify the Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant. This is known as the common difference, denoted as \(d\). To find \(d\), subtract the first term from the second term: \(d = a_{2} - a_{1} = 3 - 6 = -3\).

Step 2: Find the General Term Formula

The general term formula for an arithmetic sequence is given by \(a_{n} = a_{1} + (n-1) \, d\). Using \(a_{1} = 6\) and \(d = -3\), the formula becomes: \(a_{n} = 6 + (n-1)(-3)\). Simplify this to get: \(a_{n} = 6 - 3n + 3 = 9 - 3n\).

Step 3: Calculate \(a_{8}\)

Using the formula derived in Step 2, substitute \(n = 8\) to find \(a_{8}\): \(a_{8} = 9 - 3(8) = 9 - 24 = -15\).

Step 4: Present the Final Results

The specific term \(a_{8}\) is \(-15\). The general term for the sequence is \(a_{n} = 9 - 3n\).

Key Concepts

Common Difference

In the realm of arithmetic sequences, the common difference is a crucial concept. It's the consistent difference you observe as you move from one term to the next in the sequence.
For example, if you know two terms in the sequence, you can find this difference by subtracting the first term from the second. In our exercise, we had the terms \(a_1 = 6\) and \(a_2 = 3\). By subtracting these, \(d = a_2 - a_1 = 3 - 6 = -3\), we uncovered a common difference of \(-3\).
This value is significant as it helps us predict upcoming terms and formulate a general equation for the sequence. The key takeaway here is that finding the common difference is the first and essential step in working with arithmetic sequences.

General Term Formula

Once the common difference is identified, the next move is to construct the general term formula of the sequence. This formula lets us find any term in the sequence without listing all the preceding ones.
The formula is: \(a_n = a_1 + (n-1) \, d\), where:

• \(a_n\) is the term number \(n\) we want to find
• \(a_1\) is the first term
• \(n\) is the position of the term in the sequence
• \(d\) is the common difference

Incorporating what we've already discovered, with \(a_1 = 6\) and \(d = -3\), the formula metamorphoses into \(a_n = 6 + (n-1)(-3)\).
Simplifying gives us \(a_n = 9 - 3n\). This equation is like a magical tool, allowing us to locate any term at position \(n\) without enumerating each preceding step.

Specific Term Calculation

When you have the general term formula, you can easily find any specific term in the sequence. This is especially useful for large values, so you don't need to write out every term.
Using our derived general term formula \(a_n = 9 - 3n\), we can discover the eighth term, \(a_8\). Simply substitute \(n = 8\):

• \(a_8 = 9 - 3(8) = 9 - 24\)
• This results in \(a_8 = -15\)

Thus, \(a_8\) becomes \(-15\).
This shows how the formula works its magic, by providing specific terms directly with just a simple calculation. This step-by-step substitute approach offers clarity and precision, cementing the method's practicality for any term you aim to identify in the sequence.