Problem 21 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern of the Sequence

The given sequence is \(-2, \frac{2}{3}, -\frac{2}{9}, \frac{2}{27}, \ldots\). We must identify the common ratio, which is the factor multiplied to obtain each subsequent term from the preceding one in a geometric sequence. To do this, divide the second term by the first term: \( \frac{2}{3} \div (-2) = -\frac{1}{3} \). The common ratio is \(-\frac{1}{3}\).

Step 2: Use the Formula for the n-th term of a Geometric Sequence

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. In this sequence, \(a_1 = -2\) and \(r = -\frac{1}{3}\).

Step 3: Calculate the 7th Term

Substitute the known values into the formula: \[a_7 = -2 \cdot \left(-\frac{1}{3}\right)^{7-1} = -2 \cdot \left(-\frac{1}{3}\right)^{6}\].

Step 4: Simplify the Expression

Further simplify the calculation: \[\left(-\frac{1}{3}\right)^{6} = \frac{1}{729}\], since raising a negative number to an even power results in a positive number. Then, \[a_7 = -2 \cdot \frac{1}{729} = -\frac{2}{729}\].

Step 5: Verify the Calculation

Double-check each calculation step to ensure accuracy. The 7th term is confirmed as \(a_7 = -\frac{2}{729}\).

Key Concepts

Understanding Common Ratio in a Geometric Sequence

In the world of geometric sequences, the common ratio is a vital element. It is the constant factor by which you multiply each term to get the next term in the sequence. For instance, let's consider the sequence provided in the exercise: \(-2, \frac{2}{3}, -\frac{2}{9}, \frac{2}{27}\). To determine the common ratio (usually denoted as \(r\)), you simply divide the second term by the first term:

• The calculation is \( \frac{2}{3} \div -2 = -\frac{1}{3} \).
• This ratio, \(-\frac{1}{3}\), becomes the key to understanding the entire sequence.

Once you have the common ratio, you can predict the pattern and compute any term in the sequence. It's like having the secret code to unravel the entire sequence structure. Recognizing and calculating the common ratio is essential because it simplifies finding any term without listing all the intervening terms.

The Role of the n-th Term Formula

The n-th term formula is a powerful tool in working with geometric sequences. It allows you to find the value of any term in the sequence directly. The formula is expressed as \(a_n = a_1 \cdot r^{n-1}\), where:

• \(a_1\) is the first term of the sequence
• \(r\) is the common ratio
• \(n\) is the position of the term in the sequence

Using this formula, let's calculate the 7th term for our example sequence:

• First term \(a_1 = -2\), and the common ratio \(r = -\frac{1}{3}\).
• To find the 7th term, plug these values into the formula: \(a_7 = -2 \cdot \left(-\frac{1}{3}\right)^{7-1}\).
• This simplifies to \(-2 \cdot \left(-\frac{1}{3}\right)^6\), which equals \(-\frac{2}{729}\).

The formula makes it easy to find the value of any sequence term without manually multiplying each step. It's efficient and saves time, especially for larger values of \(n\).

Recognizing the Sequence Pattern

Identifying the sequence pattern is crucial when working with geometric sequences. A pattern in a geometric sequence is essentially a list of numbers where each term after the first is the product of the previous term and the common ratio. Observing the sequence pattern helps you predict future terms:

• In our sequence, the pattern begins with \(-2\) and follows by multiplying each term by \(-\frac{1}{3}\).
• The third term \(-\frac{2}{9}\) is found by multiplying \(\frac{2}{3}\) by \(-\frac{1}{3}\).
• This pattern continues, producing a series \(-2, \frac{2}{3}, -\frac{2}{9}, \frac{2}{27}, \ldots\).

Understanding the pattern means you don't have to rely solely on formulas; you can intuitively sense how the sequence develops. Furthermore, recognizing this sequence pattern aids in verifying your calculations, ensuring that each term follows logically from the previous term.
By mastering these patterns, you can tackle complex sequence problems with confidence.