Problem 43 - Algebra 2

Question

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 7 th term

Step-by-Step Solution

Answer

The 7th term of the given geometric sequence is 12,288.

Step 1: Identify the first term and the common ratio

By looking at the given sequence (3,12,48,192), it can be seen that the first term \(a_1\) is 3. The common ratio \(r\) is derived by dividing any term in the sequence by its preceding term. For instance, \(r = 12 / 3 = 4\).

Step 2: Substitute in the nth term formula

Substitute the identified first term, common ratio, and the term number (7) into the nth term formula. The formula becomes: \(a_7 = 3 * 4^{(7-1)}\).

Step 3: Calculate the 7th term

Calculate the expression to get the value of the 7th term. \(a_7 = 3 * 4^{6} = 3 * 4096 = 12,288\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is crucial because it determines how each term is related to the next. It is the factor that each term is multiplied by to obtain the subsequent term. To find the common ratio in a given sequence, you simply divide any term by the previous term.
For example, in the sequence \(3, 12, 48, 192, \ldots \), dividing the second term 12 by the first term 3 gives us a common ratio \(r = \frac{12}{3} = 4\).
This means each term is 4 times the term before it. The common ratio must remain consistent throughout the sequence, which defines its geometric nature. This concept is foundational as it allows for the prediction or calculation of other terms in the sequence.

The nth Term Formula of a Geometric Sequence

The ability to find any term in a geometric sequence without listing all the previous terms is a huge advantage. This is where the nth term formula comes into play. The formula is given by:

• \( a_n = a_1 \cdot r^{(n-1)} \)

where

• \(a_n\) is the nth term we're looking for,
• \(a_1\) is the first term of the sequence, and
• \(r\) is the common ratio,
• \( n \) is the term number.

Using the formula, you can find the nth term directly without needing to calculate all earlier terms, which is especially useful for large sequences.
In the earlier example, finding the 7th term involves plugging into the formula: \( a_7 = 3 \cdot 4^{(7-1)} = 3 \cdot 4^6\).
This results in the term without having to list each one until the 7th.

Sequence Calculation: Finding Specific Terms

Sequence calculation in a geometric sequence often deals with finding specific terms or patterns. Using the properties of the sequence makes this process more efficient and less error-prone.
Once we have the common ratio and the nth term formula, calculating any specific term becomes straightforward.
For example, to calculate the 7th term from the sequence \(3, 12, 48, 192, \ldots \), we determined:

• First term \(a_1 = 3\)
• Common ratio \(r = 4\)
• Term number \(n = 7\)

We apply these values using the nth term formula:

• \(a_7 = 3 \times 4^{6} = 3 \times 4096 = 12,288\)

This calculation illustrates the power of the formula in efficiently determining a specific term within the sequence. It simplifies the steps needed, saving time and ensuring the accuracy of complex sequences.