Problem 54 - College Algebra with Corequisite Support

Question

Find the \(7^{\text {th }}\) term of the geometric sequence \(\\{64 a(-b), 32 a(-3 b), 16 a(-9 b), \ldots\\}\)

Step-by-Step Solution

Answer

The 7th term is \(-729ab\).

Step 1: Identify the First Term

The first term of the geometric sequence is given as \(64a(-b)\). We can simplify this to \(-64ab\).

Step 2: Identify the Common Ratio

In a geometric sequence, the common ratio \(r\) is found by dividing the second term by the first term. The second term is \(32a(-3b)\) and the first term is \(64a(-b)\). Thus, \(r = \frac{32a(-3b)}{64a(-b)}\). Simplify this to find that \(r = \frac{-3}{2}\).

Step 3: Use the General Formula for Geometric Sequence

The formula to find the \(n^{\text{th}}\) term of a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. Here, \(a_1 = -64ab\), \(r = \frac{-3}{2}\), and \(n = 7\).

Step 4: Calculate the 7th Term

Substitute the values into the formula: \[ a_7 = -64ab \cdot \left(\frac{-3}{2}\right)^{6} \].Calculate \(\left(\frac{-3}{2}\right)^6 = \frac{729}{64}\). Then multiply:\[ a_7 = -64ab \cdot \frac{729}{64} = -729ab \].

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

The common ratio is a fundamental element of any geometric sequence. It represents the factor by which each term in the sequence is multiplied to get the subsequent term. This ratio remains constant throughout the sequence, setting the foundation for how each term relates to its neighbor.

To find the common ratio (\(r\)), one must divide the second term of the sequence by the first term. For example, in the given sequence:

• First term: \(-64ab\).
• Second term: \(32a(-3b)\).

The division of these two terms gives:\[r = \frac{32a(-3b)}{64a(-b)} = \frac{-3}{2}\].

This shows that each term in this sequence is obtained by multiplying the preceding term by \(\frac{-3}{2}\). Understanding and calculating the common ratio is crucial for grasping the behavior of geometric sequences.

Exploring the nth Term Formula for Geometric Sequences

The \(n^{\text{th}}\) term formula for a geometric sequence is the key tool for determining any specific term in the sequence. This formula is expressed as: \[a_n = a_1 \cdot r^{(n-1)}\].
Here's break down what this means:

• \(a_n\): The \(n^{\text{th}}\) term we wish to find.
• \(a_1\): The first term of the sequence. In our exercise, this is \(-64ab\).
• \(r\): The common ratio, which we previously calculated as \(\frac{-3}{2}\).
• \(n\): The position of the term we're interested in. For the exercise, we need the 7th term, so \(n = 7\).

By understanding this formula, one can predict any term within the sequence without needing to list or multiply out each term one by one. It simplifies calculations significantly and is indispensable when dealing with large sequences.

Performing Sequence Calculation with a Geometric Sequence

With an understanding of the common ratio and the \(n^{\text{th}}\) term formula, you can now calculate specific terms within a geometric sequence. Let's break down the calculation of the 7th term from our sequence:

Given:

• \(a_1 = -64ab\)
• \(r = \frac{-3}{2}\)
• \(n = 7\)

Substituting these into the formula \(a_n = a_1 \cdot r^{(n-1)}\), we find:\[a_7 = -64ab \cdot \left(\frac{-3}{2}\right)^{6}\].
First, calculate \(\left(\frac{-3}{2}\right)^6\), which simplifies to \(\frac{729}{64}\). Then multiply:\[a_7 = -64ab \cdot \frac{729}{64} = -729ab\].

This final step concludes the sequence calculation process, demonstrating how you transition from formula to specific term solution efficiently.