Problem 54 - College Algebra

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 sequence is given as \(64a(-b)\). This simplifies to \(-64ab\). Let \(a_1 = -64ab\).

Step 2: Determine the Common Ratio

The second term is \(32a(-3b)\), which simplifies to \(-96ab\). The common ratio \(r\) is calculated by dividing the second term by the first term: \(r = \frac{-96ab}{-64ab} = \frac{96}{64} = \frac{3}{2}\).

Step 3: Write the Formula for the n-th Term

The formula for the n-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 4: Substitute Values to Find the 7th Term

Substitute \(a_1 = -64ab\), \(r = \frac{3}{2}\), and \(n = 7\) into the formula: \(a_7 = -64ab \cdot \left(\frac{3}{2}\right)^{7-1}\). This simplifies to \(a_7 = -64ab \cdot \left(\frac{3}{2}\right)^6\).

Step 5: Calculate \(\left(\frac{3}{2}\right)^6\)

Calculate the value of \(\left(\frac{3}{2}\right)^6\) by computing: \(\left(\frac{3}{2}\right)^6 = \left(\frac{3^6}{2^6}\right) = \frac{729}{64}\).

Step 6: Find the 7th Term Value

Substitute back into the expression for \(a_7\): \(a_7 = -64ab \cdot \frac{729}{64}\). This simplifies to \(a_7 = -729ab\).

Key Concepts

Common Ratio

In a geometric sequence, the concept of a common ratio is crucial. It is the factor by which each term in the sequence is multiplied to get the next term. To find it, you divide any term in the sequence by the preceding term.
In the given exercise, the first term is \(-64ab\), and the second term is \(-96ab\). So, the common ratio \((r)\) is given by:

• \(r = \frac{-96ab}{-64ab} = \frac{96}{64} = \frac{3}{2}\)

Once the common ratio is identified, it can be used to predict all subsequent terms in the sequence.
Understanding the common ratio is essential because it dictates the pattern and behavior of the entire geometric sequence.

n-th Term Formula

The n-th term formula is a powerful tool in analyzing geometric sequences. This formula allows you to find any term in the sequence without having to calculate all previous terms.
In a geometric sequence, the formula for the \(n^{\text{th}}\) term is:

• \(a_n = a_1 \cdot r^{n-1}\)

Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) represents the position of the term in the sequence.
Using this formula in our exercise, we can find any term in the sequence by plugging in the known values. For example, to find the 7th term, you would use:

• \(a_7 = -64ab \cdot \left(\frac{3}{2}\right)^{7-1}\)

The n-th term formula streamlines the process of solving for any sequence term with ease.

Geometric Progression

A geometric progression is a type of sequence where each term is a fixed multiple of the previous one. This multiplier is known as the common ratio.
In the exercise, we begin with the terms \(-64ab, -96ab, \ldots\)\, which clearly form a geometric progression due to their constant ratio of \(\frac{3}{2}\).
Geometric progressions are used extensively in mathematics, finance, and science due to their predictable exponential growth or decay pattern. They provide insights into population growth, interest calculations, and more.
Recognizing a sequence as a geometric progression allows us to use specific formulas, like the n-th term formula, to efficiently find unknown terms and analyze the sequence.

Sequence Terms

Terms in a sequence are individual numbers or elements that make up the series. In a geometric sequence, each successive term is the product of the previous term and the common ratio.
In this exercise, you find the sequence terms by simplifying expressions:

• First term: \(a_1 = -64ab\)
• Second term: \(a_2 = -96ab\)
• Third term: \(a_3 = -144ab\), and so on.

In order to get to the 7th term, we leverage the sequence's regularity and use the n-th term formula.
Each term has a specific place and value, guided by the initial term and the common ratio, forming the unique essence of a geometric sequence. Understanding each term's formation helps in predicting future terms and recognizing the overall pattern of the sequence.