Problem 52 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the geometric sequence. $$a_{1}=0.8, r=-3$$

Step-by-Step Solution

Answer

The general term is \(a_n = 0.8 \, (-3)^{n-1}\).

Step 1: Understand Geometric Sequence Formula

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

Step 2: Identify First Term and Common Ratio

We are given \(a_1 = 0.8\) as the first term of the sequence and \(r = -3\) as the common ratio.

Step 3: Substitute Known Values into the Formula

Substitute the values \(a_1 = 0.8\) and \(r = -3\) into the formula for the general term: \(a_n = 0.8 \, (-3)^{n-1}\).

Step 4: Write the General Term

The general term of the geometric sequence is \(a_n = 0.8 \, (-3)^{n-1}\).

Key Concepts

General Term

In the context of geometric sequences, the general term is crucial because it allows us to find any term in the sequence without calculating all preceding terms. A general term is typically denoted as \(a_n\), where \(n\) represents the specific position in the sequence. Rather than manually multiplying the terms repeatedly, the general term formula does the heavy lifting.

For a geometric sequence, the general term is given by:

• \(a_n = a_1 \cdot r^{n-1}\)

Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position number. This formula is handy because it simplifies the process of finding any term in the sequence. Just plug in the values for the first term, the common ratio, and the desired term position \(n\).

Understanding the general term can enhance your grasp of geometric sequences and solve related problems efficiently.

Common Ratio

The common ratio is a key component in determining the progression of a geometric sequence. It's the consistent factor by which each term is multiplied to get to the next term. In our example, this commonality helps keep the sequence uniform across its entirety.

To determine the common ratio \(r\), observe the ratio between any pair of consecutive terms. Mathematically, it's expressed as the quotient of two successive terms:

• \(r = \frac{a_{2}}{a_{1}}\)

This constant ratio is critical as it's applied repeatedly across the sequence. In the original exercise, the common ratio \(r\) is given as \(-3\). This means each term is obtained by multiplying the previous term by \(-3\).

The knowledge of the common ratio not only helps you predict future terms but also offers insight into the sequence's behavior. For instance, if \(r\) is negative, the terms will alternate in sign, flipping positive to negative and vice versa.

Sequence Formula

The sequence formula for a geometric sequence is like a blueprint, providing a systematic way to construct and understand the sequence. This formula serves as a compelling example of mathematical elegance in how a simple expression can describe complex patterns.

The standard sequence formula for a geometric sequence is:

• \(a_n = a_1 \cdot r^{n-1}\)

Here, \(a_1\) is the initial term, \(r\) is the common ratio, and \(n\) is the term number. Using this formula, you can easily find the value of any term in the sequence by just substituting the appropriate \(n\).

For the sequence discussed in the exercise, the formula becomes \(a_n = 0.8 \cdot (-3)^{n-1}\). This equation empowers you to predict any term’s value by simply changing \(n\). Understanding this formula deepens your mathematical insight and is a powerful tool in solving related geometric sequence problems efficiently.