Problem 25 - Beginning and Intermediate Algebra

Question

Find a formula for the general term, \(a_{n},\) of each sequence. $$\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \dots$$

Step-by-Step Solution

Answer

The general term, \(a_n\), of the geometric sequence \(\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \dots\) is given by the formula: \[a_n = \frac{1}{3} * \left(\frac{1}{3}\right)^{n-1}\]

Step 1: Identify the common ratio and sequence type

First, let's observe the given terms of the sequence: \(\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \dots\). Notice that, in order to obtain the next term, we multiply the previous term by the same value each time, which is 1/3. For example, \(\frac{1}{3} * \frac{1}{3} = \frac{1}{9}\), \(\frac{1}{9} * \frac{1}{3} = \frac{1}{27}\), and so on(\$a_1 = \frac{1}{3}\). Since there is a common ratio between consecutive terms, we know it is a geometric sequence.

Step 2: Apply the geometric sequence formula for the general term

Now that we know it is a geometric sequence, we can use the formula for the general term of a geometric sequence, which is given by: \[a_n = a_1 * r^{n-1}\] Here, \(a_n\) represents the nth term of the sequence, \(a_1\) is the first term of the sequence, \(r\) is the common ratio between consecutive terms, and \(n\) is the position number of the term in the sequence.

Step 3: Calculate the general term for the given sequence

In our case, the first term, \(a_1\), is \(\frac{1}{3}\), and the common ratio, \(r\), is also \(\frac{1}{3}\). Plugging these values into the formula, we get: \[a_n = \frac{1}{3} * \left(\frac{1}{3}\right)^{n-1}\] This is the formula for the general term of the given sequence.

Key Concepts

General Term Formula

In a geometric sequence, finding the general term formula is a crucial skill. This formula allows us to predict any term's value based on its position in the sequence. In a geometric sequence, the formula for the general term, denoted as \( a_n \), can be found using:\[ a_n = a_1 \cdot r^{n-1} \] where:

• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio (the factor by which we multiply each term to get the next),
• \( n \) is the term number.

By using this formula, we can calculate any term in the sequence without listing all previous terms. For example, given our sequence, where the first term \( a_1 \) is \( \frac{1}{3} \) and the common ratio \( r \) is also \( \frac{1}{3} \), we can determine any term \( a_n \) by plugging these values into the formula.

Common Ratio

The common ratio is an essential element of a geometric sequence. It determines how each term is related to the previous one. In our example sequence \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \dots \), the common ratio \( r \) is \( \frac{1}{3} \). This means each term is obtained by multiplying the previous term by \( \frac{1}{3} \).

To find the common ratio in any geometric sequence, divide a term by its preceding term. For instance:

• \( \frac{1}{9} \div \frac{1}{3} = \frac{1}{3} \)
• \( \frac{1}{27} \div \frac{1}{9} = \frac{1}{3} \)

Knowing the common ratio helps in writing the general term formula and understanding how the sequence progresses.

Sequence Types

Sequences come in different types, and recognizing the type is the first step in solving related problems. Two primary types of sequences are arithmetic and geometric sequences. Our focus is on geometric sequences:

• Arithmetic Sequence: Each term is obtained by adding a constant value called the "common difference" to the previous term.
• Geometric Sequence: Each term is obtained by multiplying the previous term by a constant value, known as the "common ratio." This is the type of sequence we are working with.

In the given sequence \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \dots \), the defining feature is the multiplication of terms by the common ratio \( \frac{1}{3} \). Understanding the specific type of sequence allows us to apply the correct formula to find the general term or any specific term in the sequence.