Problem 54 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the geometric sequence. $$a_{4}=3, r=3$$

Step-by-Step Solution

Answer

The general term is \( a_{n} = \frac{1}{9} \, 3^{n-1} \).

Step 1: Understanding the Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio, denoted as \( r \). Given that \( r = 3 \), this is our ratio for this problem.

Step 2: Define the General Formula

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

Step 3: Express the Fourth Term

Using the general formula for the fourth term \( a_4 \), we have:\[a_4 = a_1 \, r^{3}\] Given that \( a_4 = 3 \) and \( r = 3 \), substitute these values to get:\[3 = a_1 \, 3^3\]

Step 4: Solve for the First Term

Simplify \( 3^3 \) to \( 27 \), and then solve for \( a_1 \):\[3 = a_1 \, 27\]Divide both sides by 27:\[a_1 = \frac{3}{27} = \frac{1}{9}\]

Step 5: Substitute Back to Find General Term

Now that we have \( a_1 = \frac{1}{9} \), substitute back into the general formula to find \( a_n \):\[a_n = \frac{1}{9} \, 3^{n-1}\] This is the expression for the general term of the sequence.

Key Concepts

General Term Formula

The general term formula helps us determine any term in a geometric sequence without listing all previous terms. This formula is essential in understanding how geometric sequences grow or shrink.
For a geometric sequence, the formula for the general term, often represented as \( a_n \), is:

• \( a_n = a_1 \cdot r^{n-1} \)

Here, \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the term number. Using this formula, we can find the value of any term in the sequence if we know the first term and the common ratio.
In our exercise, we identified the fourth term \( a_4 \) as 3, along with the common ratio \( r \) being 3. Our task was to find this first term in order to write a complete formula for \( a_n \). The process involved comparing the given term to the formula and solving for \( a_1 \).

Common Ratio

The common ratio is a crucial element in geometric sequences. It is the fixed number we multiply each term by to get the next term, and it serves as the backbone for the sequence's pattern.
To identify a common ratio in a sequence, you can divide any term by the term before it. In our exercise, we are told that \( r = 3 \), which means each term is three times bigger than the one before it.
Knowing the common ratio allows for quick calculations and helps maintain the sequence’s uniformity. For example, by multiplying the first term by \( r \) repeatedly, you can easily find successive terms:

• If \( a_1 = \frac{1}{9} \), then \( a_2 = \frac{1}{9} \cdot 3 = \frac{1}{3} \).
• Consequently, \( a_3 = \frac{1}{3} \cdot 3 = 1 \), and so forth.

This understanding is key to accurately setting up and solving problems involving geometric sequences.

Sequence Terms

Sequence terms in a geometric sequence are the individual elements ordered in a specific pattern based on the common ratio. Each term builds on the last one, creating a structured series.
Let's look at how knowing terms is significant. With our general term formula established as \( a_n = \frac{1}{9} \cdot 3^{n-1} \), each term can be easily calculated. For instance:

• The first term \( a_1 = \frac{1}{9} \)
• The second term \( a_2 = \frac{1}{9} \cdot 3 = \frac{1}{3} \)
• The third term \( a_3 = \frac{1}{3} \cdot 3 = 1 \)
• The fourth term \( a_4 = 1 \cdot 3 = 3 \)

Each step involves multiplying the previous term by the common ratio \( r \). Sequence terms are vital in understanding the behavior of the sequence and finding patterns or predicting future terms efficiently. By learning how each term evolves from its predecessor using the ratio, one can appreciate the sequence's consistency and prediction power.