Problem 68 - College Algebra with Corequisite Support

Question

Find the \(5^{\text {th }}\) term of the arithmetic sequence \(\\{9 b, 5 b, b, \ldots\\}\).

Step-by-Step Solution

Answer

The 5th term is \(-7b\).

Step 1: Identify the Sequence Terms

The given sequence starts with the terms \(9b, 5b, b, \ldots\). Let's denote the first term (\(a_1\)) as \(9b\) and the second term (\(a_2\)) as \(5b\).

Step 2: Find the Common Difference

The common difference \(d\) in an arithmetic sequence is found by subtracting the first term from the second term. Thus, \(d = a_2 - a_1 = 5b - 9b = -4b\).

Step 3: Use the Arithmetic Sequence Formula

The formula to find the \(n^{\text{th}}\) term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). We want to find the 5th term (\(a_5\)).

Step 4: Calculate the 5th Term

Substitute \(a_1 = 9b\), \(d = -4b\), and \(n = 5\) into the formula: \[a_5 = 9b + (5-1) \cdot (-4b) \]. Simplifying this provides: \[a_5 = 9b - 16b = -7b\].

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key feature that sets the pattern between consecutive terms. It can be thought of as the consistent "step size" that you repeatedly add (or subtract) to progress from one term to the next. Finding the common difference requires you to subtract the first term from the second.
For instance, in the given sequence with terms \(9b, 5b, b, \ldots\), the calculation goes like this:

• First term, \(a_1 = 9b\)
• Second term, \(a_2 = 5b\)

The common difference \(d\) is determined by:

• \(d = a_2 - a_1 = 5b - 9b = -4b\)

Thus, each subsequent term decreases by \(-4b\) from its predecessor. This makes it easy to predict all other terms in the sequence just by continuing to subtract \(-4b\). Knowing this difference helps us quickly move from one term to the next in the sequence.

Arithmetic Sequence Formula

To efficiently determine any term in an arithmetic sequence, we make use of the arithmetic sequence formula. This formula plays a pivotal role as it links the sequence’s first term, the common difference, and any desired term position all together. Understanding this formula can help simplify your calculations greatly.
The standard formula to find the \(n^{\text{th}}\) term, \(a_n\), is written as:

• \(a_n = a_1 + (n-1) \cdot d\)

Here’s how each component matters:

• \(a_1\) is your initial term in the sequence
• \((n-1)\) indicates how many steps away you are from the first term
• \(d\) is the common difference

By framing problem-solving within this formula, you can locate any term without manually progressing through each one. That’s particularly useful for sequences with large numbers of terms!

Term Calculation

When tasked with finding a specific term within an arithmetic sequence, applying the arithmetic sequence formula simplifies the process. Let's walk through calculating the 5th term as in our exercise.
First, identify what you already know:

• First term \(a_1 = 9b\)
• Common difference \(d = -4b\)
• Desired term position \(n = 5\)

Plugging these into the formula:

• \(a_5 = a_1 + (n-1) \cdot d\)
• \(a_5 = 9b + (5-1) \cdot (-4b)\)
• \(a_5 = 9b - 16b = -7b\)

Therefore, the 5th term of the sequence is \(-7b\). By understanding and using the formula effectively, you can master quick calculation of any term, regardless of its position in the sequence.