Problem 44 - Algebra 2

Question

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 10 th term

Step-by-Step Solution

Answer

The 10th term in the given geometric sequence is 196608.

Step 1: Identify the Common Ratio

Observe the given sequence \(3,12,48,192, \ldots,\), it can be identified that the common ratio, often denoted by 'r', is 4, as each term is four times the previous term.

Step 2: Apply Geometric Sequence Formula

The nth term of a geometric sequence is given by the formula \(a_n = a_1 * r^{(n-1)}\), where \(a_n\) is the nth term, \(a_1\) is the first term, and \(r\) is the common ratio. Here, \(a_1\) = 3, \(r\) = 4, and \(n\) = 10.

Step 3: Compute the 10th Term

Applying the formula, the 10th term = \(3 * 4^{(10-1)}\). After performing the calculation, the 10th term is found to be 196608

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key concept that determines how the sequence progresses. It is the factor by which each term in the sequence is multiplied to obtain the next term.
To find the common ratio, simply divide any term in the sequence by the previous term. For example, in the sequence provided: \(3, 12, 48, 192, \ldots\), dividing the second term (12) by the first term (3) gives us \(\frac{12}{3} = 4\). This tells us that the common ratio is 4.

• The common ratio is constant throughout the sequence.
• A sequence with a common ratio greater than 1 will grow, while a common ratio between 0 and 1 will lead to a decreasing sequence.

Understanding the common ratio helps us predict future terms and grasp the behavior of the entire sequence.

Nth Term Formula

The nth term formula is a powerful tool that allows us to find any term in a geometric sequence without having to list all the previous terms.
The formula is expressed as \(a_n = a_1 \cdot r^{(n-1)}\) where:

• \(a_n\) is the nth term.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number you want to find.

Using this formula, you can quickly compute any term in the sequence. For instance, using \(a_1 = 3\), \(r = 4\), and \(n = 10\), the 10th term can be calculated as:
\(a_{10} = 3 \cdot 4^{9} = 196608\).
This shows how the formula simplifies finding specific terms, especially far along in the sequence.

Geometric Progression

Geometric progression refers to a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number, known as the common ratio.
This sequence is characterized by a constant multiplication factor between consecutive terms.

• An example includes the sequence \(3, 12, 48, 192,\ldots\), where each term is four times the previous term.
• Geometric sequences can be increasing, as in the given example, or decreasing if the common ratio is a fraction.
• The series can also be infinite or finite, depending on how it's defined.

Geometric progressions are useful in various real-life applications, such as calculating compound interest, analyzing population growth, and forecasting financial trends. Understanding how geometric progressions work enables us to model scenarios and make predictions based on a constant rate of change.