Problem 39 - College Algebra with Corequisite Support

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

Step 1: Identify the First Term

The first term of the sequence is given as \( a_1 = -1 \). This is the starting term from which the geometric sequence progresses.

Step 2: Determine the Common Ratio

The common ratio \( r \) in a geometric sequence can be found by dividing the second term by the first term. Calculate \( r = \frac{-\frac{4}{5}}{-1} = \frac{4}{5} \).

Step 3: Write the Explicit Formula

The formula for the \( n \)-th term of a geometric sequence is given by \( a_n = a_1 \cdot r^{n-1} \). Substitute \( a_1 = -1 \) and \( r = \frac{4}{5} \) into the formula to get \( a_n = -1 \cdot \left( \frac{4}{5} \right)^{n-1} \).

Step 4: Simplify the Explicit Formula

Simplify the formula by multiplying out the expression if necessary. Here, \( a_n = - \left( \frac{4}{5} \right)^{n-1} \) is already in its simplest form.

Key Concepts

Explicit Formula

The explicit formula is a key concept in understanding sequences, especially geometric ones. It allows you to determine any term in the sequence without having to list out all the previous terms. The general formula for the n-th term of a geometric sequence is:

• \( a_n = a_1 \cdot r^{n-1} \)

In this formula, \( a_1 \) represents the first term, \( r \) is the common ratio, and \( n \) is the term number you want to find. This structure quickly reveals the value of any term based on its position. For our sequence, we found that \( a_1 = -1 \) and \( r = \frac{4}{5} \), so the explicit formula is:

• \( a_n = -1 \cdot \left( \frac{4}{5} \right)^{n-1} \)

This formula is not only simple but powerful, allowing us to calculate any term in the sequence with ease. Just plug in the value of \( n \) to find your desired term.

Common Ratio

The common ratio plays a crucial role in geometric sequences. It defines the factor by which we multiply a term to get the next term in the progression. To find the common ratio in any geometric sequence, divide any term by the previous term.Here, we calculated the common ratio \( r \) by dividing the second term by the first term:

• \( r = \frac{-\frac{4}{5}}{-1} = \frac{4}{5} \)

Notice that the sign of the common ratio matters in the context of the original sequence. Since both terms were negative, the ratio is positive. This positive common ratio indicates that subsequent terms keep getting closer to zero but remain negative, following the same pattern of progression as we move through the sequence. Understanding the common ratio helps in predicting the behavior of the sequence, ensuring accuracy in calculations.

Geometric Progression

The term geometric progression refers to a sequence where each term is obtained by multiplying the previous term by a constant, known as the common ratio. Each term, therefore, follows a specific pattern, making it easy to predict future terms if you know the progression:

• First, recognize the starting term, \( a_1 \).
• Second, determine the common ratio, \( r \).
• Use the explicit formula to find any term.

In our example, the sequence is given as \(-1, -\frac{4}{5}, -\frac{16}{25}, \ldots\) and follows a geometric progression with \( a_1 = -1 \) and \( r = \frac{4}{5} \). The sequence diminishes progressively closer to zero, reflecting how geometric sequences can be used in modeling situations where values decay or shrink over time. Recognizing a geometric progression helps in understanding trends and predicting patterns in various fields, from finance to natural sciences.