Problem 14 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of a geometric sequence. If the sequence is geometric, find the common ratio. $$3,48,93,138, \dots$$

Step-by-Step Solution

Answer

The sequence is not geometric because the ratios are not consistent.

Step 1: Identify Formula for Geometric Sequence

In a geometric sequence, each term is obtained by multiplying the previous term by a constant known as the common ratio, denoted as \( r \). The formula for the nth term of a geometric sequence is given by \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term.

Step 2: Calculate Ratios Between Consecutive Terms

To determine if the sequence is geometric, calculate the ratio between each pair of consecutive terms: \( \frac{48}{3} \), \( \frac{93}{48} \), and \( \frac{138}{93} \).

Step 3: Compute Each Ratio

Perform the calculations: \( \frac{48}{3} = 16 \), \( \frac{93}{48} = 1.9375 \), and \( \frac{138}{93} \approx 1.4839 \).

Step 4: Compare Ratios

Compare the ratios: the ratios \( 16 \), \( 1.9375 \), and \( 1.4839 \) are not equal. Therefore, the sequence is not geometric as the ratios are not constant.

Key Concepts

Sequence Analysis

When analyzing a sequence, it's important to understand what type of sequence you are dealing with, as each type follows its own specific rules. In the case of geometric sequences, the focus is on multiplication by a constant factor. Each term is created by multiplying the previous term by a fixed amount known as the common ratio. This characteristic is what sets geometric sequences apart from other types, such as arithmetic sequences, which involve addition or subtraction by a constant amount.

In the exercise provided, we are given the first four terms of a sequence: 3, 48, 93, and 138. The task is to determine whether these form a geometric sequence. To do this, we need to thoroughly analyze the relationships between these numbers. Understanding whether a sequence is geometric can help predict future terms and find solutions to real-life problems involving growth or decay.

Common Ratio

The common ratio is key in determining whether a sequence is geometric. It is the constant factor you multiply each term by to get the next term. In a geometric sequence, the ratio, denoted as \( r \), should remain the same between each pair of consecutive terms.

To find the common ratio in the given sequence (3, 48, 93, 138), one needs to divide each term by the preceding term:

• First, calculate \( \frac{48}{3} = 16 \)
• Then, \( \frac{93}{48} = 1.9375 \)
• Finally, \( \frac{138}{93} \approx 1.4839 \)

As evident from these calculations, the ratios are not consistent. For a sequence to be geometric, all of these ratios would need to be equal. Because they are different, the sequence in question cannot be classified as a geometric sequence. Thus, in this case, there is no common ratio.

Geometric Progression Formula

In mathematics, the geometric progression formula helps in identifying the nth term of a geometric sequence. This is especially useful for finding specific terms without listing all the preceding ones. The formula is expressed as:

• \( a_n = a_1 \cdot r^{n-1} \)

Here, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence.

Using this formula, if a sequence is indeed geometric, you can easily calculate any term in the sequence. However, since the given sequence does not have a constant common ratio, this formula is not applicable here. For sequences that adhere to the principles of geometric progression, this formula is a powerful tool, allowing predictions of future terms accurately and efficiently.