Problem 39 - College Algebra

Question

For the following exercises, write an explicit formula for each geometric sequence. $$ a_{n}=\left\\{-1,-\frac{4}{5},-\frac{16}{25},-\frac{64}{125}, \ldots\right\\} $$

Step-by-Step Solution

Answer

The explicit formula is \(a_n = -1 \left(\frac{4}{5}\right)^{n-1}\).

Step 1: Identify the first term

The sequence provided begins with the term \(-1\). Therefore, the first term \(a_1\) is \(-1\).

Step 2: Determine the common ratio

To find the common ratio \(r\), we take the ratio of the second term to the first term. The second term is \(-\frac{4}{5}\) and the first term is \(-1\). Thus, \(r = \frac{-\frac{4}{5}}{-1} = \frac{4}{5}\).

Step 3: Write the explicit formula

The formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \, r^{n-1}\). Using \(a_1 = -1\) and \(r = \frac{4}{5}\), the explicit formula becomes \(a_n = -1 \left(\frac{4}{5}\right)^{n-1}\).

Key Concepts

Explicit Formula

In a geometric sequence, the explicit formula is a way to calculate any term in the sequence without needing the previous term. Rather than calculating each term one by one, the explicit formula provides a direct link between any term number \(n\) and its value. The general form of the explicit formula for a geometric sequence is given by:

• \( a_n = a_1 \, r^{n-1} \)

Here, \(a_1\) is the first term of the sequence, \(r\) is the common ratio, and \(n\) is the term number. This formula is particularly useful because it allows you to quickly find any term in the sequence by substituting the appropriate values for \(a_1\), \(r\), and \(n\). Even if \(n\) is very large, this formula provides a simple and efficient way to calculate \(a_n\) directly based on its position in the sequence. It highlights the power and simplicity of the geometric sequence formula.

Common Ratio

The common ratio is a central concept in geometric sequences and refers to the factor by which we multiply each term to find the next term. In a given geometric sequence, this ratio stays constant, which is what defines the sequence's regular growth or decay pattern. To determine the common ratio \(r\) in a sequence, we divide any term by the previous term. For example, in the sequence \(-1, -\frac{4}{5}, -\frac{16}{25}\), the common ratio is found by dividing \(-\frac{4}{5}\) by \(-1\), giving \(\frac{4}{5}\).

• This shows that each term is \(\frac{4}{5}\) of the preceding term.

Knowing the common ratio is crucial because it directly influences the shape and behavior of the sequence. If \(|r| < 1\), the sequence will gradually decrease in size, converging towards zero. If \(|r| > 1\), the terms will grow indefinitely. Understanding the common ratio also ensures you can accurately apply the explicit formula for geometric sequences.

First Term

The first term in a geometric sequence is denoted as \(a_1\) and serves as the starting point of the sequence. It acts as the "anchor" from which all other terms are derived using the common ratio. When given a sequence, identifying the first term is essential because it sets the initial value from which the geometric progression starts. For instance, in the sequence provided, the first term is \(-1\), which means the sequence begins with \(-1\).

• This first term plays a pivotal role in the sequence's explicit formula \(a_n = a_1 \, r^{n-1}\).

By understanding and identifying \(a_1\), we can ensure that our calculations using the explicit formula start accurately. The first term doesn't just initiate the sequence but also determines its overall values when combined with the common ratio in subsequent terms. Recognizing and using the first term appropriately is crucial in solving problems involving geometric sequences effectively.