Problem 28 - Beginning and Intermediate Algebra

Question

Find the general term of each geometric sequence. $$2, \frac{2}{3}, \frac{2}{9}, \frac{2}{27}, \dots$$

Step-by-Step Solution

Answer

The general term of the geometric sequence is \(ar^{n-1}=2 \cdot \left(\frac{1}{3}\right)^{n-1}\).

Step 1: Identify the first term

The first term in the sequence is 2.

Step 2: Find the common ratio

To find the common ratio, divide the second term by the first term. In this case, divide \(\frac{2}{3}\) by 2: \[\frac{\frac{2}{3}}{2} = \frac{2}{3} \cdot \frac{1}{2} = \frac{2}{6} = \frac{1}{3}\] So, the common ratio is \(\frac{1}{3}\).

Step 3: Write the general term of the sequence

Now that we have the first term and the common ratio, we can write the general term of the sequence as \(ar^{n-1}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term. In this case, the first term \(a = 2\) and the common ratio \(r =\frac{1}{3}\). Therefore, the general term is: \[ar^{n-1}=2 \cdot \left(\frac{1}{3}\right)^{n-1}\]

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the 'common ratio' is the factor that determines what each term will be multiplied by to get the next term. This makes it an essential component of geometric sequences. When you have a sequence, like the one given: 2, \(\frac{2}{3}\), \(\frac{2}{9}\), \(\frac{2}{27}\), etc., you can find the common ratio by dividing one term by the preceding term.

In our example, to find the common ratio, you start with the second term \(\frac{2}{3}\) and divide it by the first term 2:

• \(\frac{2}{3} \div 2 = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3}\)

This means that each term is \(\frac{1}{3}\) of the previous term. Recognizing and calculating the common ratio correctly is vital since this affects how the entire sequence behaves.

Writing the General Term Formula

The general term formula provides a way to find any term in the sequence without having to list all the previous terms. For any geometric sequence, the general term can be determined by the formula \(a r^{n-1}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

In our sequence, we have:

• First term \(a = 2\)
• Common ratio \(r = \frac{1}{3}\)

Thus, the general term for this sequence will be:

• \(a r^{n-1} = 2 \left(\frac{1}{3}\right)^{n-1}\)

This formula helps you instantly find the value of any term in the sequence without calculating each previous term, which is especially useful for large sequences or finding terms far apart in the sequence.

First Term Identification

Knowing the first term of a geometric sequence is crucial as it serves as the starting point. The first term is usually denoted as \(a\). It's the groundwork for applying the general term formula and finding subsequent terms.

In the sequence given: 2, \(\frac{2}{3}\), \(\frac{2}{9}\), \(\frac{2}{27}\), the first term is 2. This means:

• \(a = 2\)

Establishing the first term correctly ensures that the entire sequence aligns properly when using the general term formula. It's like having the right seed planted for the rest of the sequence to grow correctly. Understanding this connection will help you approach geometric sequences with more confidence and clarity.