Problem 17 - College Algebra

Question

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\)to find \(a_{7},\) the seventh term of the sequence. $$3,12,48,192, \dots$$

Step-by-Step Solution

Answer

The general term for this geometric sequence can be represented as \(a_n = 3 \times 4^{n-1}\). When n equals 7, the seventh term, \(a_7\), equals 12288.

Step 1: Find the Common Ratio

First, we have to identify the common ratio of the geometric sequence. Divide the second term by the first term, i.e, \(r = 12/3 = 4\). The common ratio (\(r\)) is 4.

Step 2: Formulate the General Form

Now, we can define a general formula based on the general term of a geometric sequence. We know the first term is \(a = 3\), and we just found that \(r = 4\). So the general form is \(a_n = 3 \times 4^{(n-1)}\)

Step 3: Find the Seventh Term

To find the seventh term, or \(a_7\), substitute \(n = 7\) in your newly found formula. Therefore, \(a_7 = 3 \times 4^{(7-1)} = 3 \times 4^6\), which equals 12288 when computed.

Key Concepts

Common Ratio

Every geometric sequence has a property known as the common ratio. To find this, you divide any term in the sequence by the term that comes before it. In our sequence, which begins with 3, 12, 48, 192, ..., the common ratio is calculated by dividing 12 by 3, giving us 4. This means that each term is 4 times the previous term. Understanding the common ratio is crucial because it determines how the sequence progresses. It also plays a fundamental role in deriving the general term formula, which connects the terms of a sequence.

General Term Formula

The general term formula of a geometric sequence allows us to generate any term of the sequence without listing all preceding terms. The formula is given by \(a_n = a \times r^{(n-1)}\), where \(a\) is the first term and \(r\) is the common ratio. In our example:

• First term \(a = 3\)
• Common ratio \(r = 4\)

Substituting these values into the formula, we obtain \(a_n = 3 \times 4^{(n-1)}\). This compact formula saves time and effort and is especially useful when we need to find terms far along in the sequence.

Nth Term

In the context of geometric sequences, the term \(n\) typically represents the position of a term in the sequence. The formula \(a_n = a \times r^{(n-1)}\) lets us find the \(n\)th term directly. This means, given any position \(n\), you can quickly determine the term that occupies that position without having to calculate all the preceding terms. This formula is derived from the basic structure of a geometric sequence, leveraging the common ratio to understand how each subsequent term is derived from the first.

Seventh Term

Once we have the general term formula, finding the seventh term becomes simple. We substitute \(n = 7\) into our formula \(a_n = 3 \times 4^{(n-1)}\). This gives us \(a_7 = 3 \times 4^{6}\). Calculating this, we find \(a_7 = 3 \times 4096\), which equals 12288. This process highlights how powerful the general term formula is—it allows us to compute terms deep in the sequence with minimal calculation, bypassing the need to find every term leading up to the seventh.