Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\left\\{2, \frac{1}{3}, \frac{1}{18}, \frac{1}{108}, \ldots\right\\}\)

Step-by-Step Solution

Answer

The explicit formula is \(a_n = 2 \cdot \left(\frac{1}{6}\right)^{n-1}\).

Step 1: Identify the First Term

In a geometric sequence, the first term, denoted as \(a_1\), is the starting point of the sequence. For this sequence, the first term \(a_1\) is given as \(2\).

Step 2: Determine the Common Ratio

The common ratio \(r\) is found by dividing the second term of the sequence by the first term. Here, \(r = \frac{1/3}{2} = \frac{1}{6}\).

Step 3: Write the Explicit Formula for a Geometric Sequence

The explicit formula for the \(n\)-th term \(a_n\) of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\). Substitute \(a_1 = 2\) and \(r = \frac{1}{6}\) into the formula.

Step 4: Substitute the Values

Plug the values \(a_1 = 2\) and \(r = \frac{1}{6}\) into the explicit formula. This results in \(a_n = 2 \cdot \left(\frac{1}{6}\right)^{n-1}\).

Step 5: Simplify the Formula (if necessary)

The formula is already in its simplest form: \(a_n = 2 \cdot \left(\frac{1}{6}\right)^{n-1}\). There is no further simplification required.

Key Concepts

Explicit Formula

In the world of geometric sequences, the explicit formula is a powerful tool. It allows you to quickly find any term in the sequence without having to write out all the preceding terms. For a geometric sequence, the explicit formula is generally written as \(a_n = a_1 \cdot r^{n-1}\). Here:

• \(a_n\) represents the \(n\)-th term you wish to find,
• \(a_1\) is the first term in the sequence,
• \(r\) is the common ratio, which is the factor you multiply by to get from one term to the next,
• \(n\) is the term number you're interested in finding.

To use the formula, substitute the known values for \(a_1\), \(r\), and \(n\) into the equation, and you'll obtain the explicit value for \(a_n\). This makes it super handy for calculating any term directly.

First Term

The journey of a geometric sequence begins with its first term, denoted as \(a_1\). This initial term sets the stage for the entire sequence and is crucial for determining the explicit formula. In our example, we start with \(a_1 = 2\). Starting with the correct first term ensures that every subsequent term falls neatly into place according to the common ratio. When dealing with sequences, knowing the first term helps define the entire path the sequence will follow.

Common Ratio

A geometric sequence is characterized by its common ratio, denoted as \(r\). This is the constant factor by which each term is multiplied to get the next term. To find the common ratio, you divide the second term by the first term. For example, with the sequence \( \left\{2, \frac{1}{3}, \ldots\right\} \), the common ratio \(r\) was calculated as \(\frac{1/3}{2} = \frac{1}{6}\).

Knowing the common ratio is essential because it helps maintain the consistent relationship between terms, allowing the sequence to progress smoothly. Once you've calculated \(r\), you can predict the behavior and growth pattern of the sequence by multiplying it continuously.

Sequence

Geometric sequences are a type of sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. Sequences follow a specific order of numbers, and for a geometric sequence, this order is determined by the common ratio. In our exercise, the sequence starts with \(\{2, \frac{1}{3}, \frac{1}{18}, \ldots\}\).

The structure of a geometric sequence is governed by its explicit formula, which encapsulates all necessary information like the first term and the common ratio. Once you understand how the sequence works, using the explicit formula becomes a breeze for identifying any term within the sequence.