Problem 21 - College Algebra

Question

For the following exercises, find the specified term for the geometric sequence, given the first four terms. $$a_{n}=\left\\{-2, \frac{2}{3},-\frac{2}{9}, \frac{2}{27}, \ldots .\right\\} \text { Find } a_{7}$$

Step-by-Step Solution

Answer

The 7th term is \(-\frac{2}{729}\).

Step 1: Identify the sequence type

The given sequence is geometric because it is characterized by a constant ratio between successive terms.

Step 2: Find the common ratio

To find the common ratio, divide the second term by the first term: \(r = \frac{\frac{2}{3}}{-2} = -\frac{1}{3}\).

Step 3: Use the formula for the nth term

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

Step 4: Identify the first term

From the sequence, \(a_1 = -2\).

Step 5: Calculate the 7th term

Substitute \(a_1 = -2\), \(r = -\frac{1}{3}\), and \(n = 7\) into the formula: \[a_7 = -2 \left(-\frac{1}{3}\right)^{6} = -2 \times \frac{1}{729} = -\frac{2}{729}.\]

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a critical component. It represents the factor by which we multiply one term to get to the next term. Identifying the common ratio helps us understand how the sequence progresses.
To find it:

• Pick any two consecutive terms in the sequence. For example, from the sequence \(-2, \frac{2}{3}, -\frac{2}{9}, \frac{2}{27}\) we can choose the first and second terms.
• Divide the second term by the first term: \(r = \frac{\frac{2}{3}}{-2} = -\frac{1}{3}\).

Once calculated, this ratio (\(-\frac{1}{3}\) in our case) remains the same throughout the sequence. This constancy is what defines a geometric sequence. Understanding and calculating the common ratio allows you to determine any subsequent term in the sequence easily.

Nth Term Formula

The nth term formula is a powerful tool for finding any term in a geometric sequence. The formula is expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where:

• \(a_n\) is the term we want to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the position number of the term in question.

For example, if we want to calculate the 7th term of our sequence, we substitute \(a_1 = -2\), \(r = -\frac{1}{3}\), and \(n = 7\) into the formula:
\[ a_7 = -2 \left(-\frac{1}{3}\right)^{6} = -2 \times \frac{1}{729} = -\frac{2}{729} \]
The nth term formula gives us a direct way to compute any term in the sequence without having to list all the preceding terms. This makes it efficient and especially useful for sequences with long indices.

Geometric Sequence Problems

Geometric sequence problems often involve determining specific terms in the sequence or finding missing values. When faced with these problems, the first step is always to identify the common ratio and the first term. This foundational information is essential for using the nth term formula.
Some typical problems might include:

• Finding a specified term: Like calculating \(a_7\) as in our example sequence.
• Determining if a given sequence is geometric: By checking if a constant ratio exists.
• Solving for unknowns: If parts of the sequence or its terms are missing, use the formula to solve for these unknowns.

Understanding the common ratio and the application of the nth term formula can simplify solving these problems. Each step in solving geometric sequence problems builds upon your understanding of these key elements. With practice, identifying patterns and using formulas will become intuitive, aiding you in tackling more complex problems efficiently.