Problem 9 - College Algebra

Question

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. $$ -6,-12,-24,-48,-96, \ldots $$

Step-by-Step Solution

Answer

Yes, the sequence is geometric. The common ratio is 2.

Step 1: Understanding Geometric Sequences

A sequence is geometric if the ratio between consecutive terms is constant. This constant ratio is called the 'common ratio'. We need to determine if the given sequence has this property.

Step 2: Calculate Consecutive Ratios

To find the common ratio, we divide each term by the previous one. Calculate the following ratios for the sequence: \(-12 \div -6\), \(-24 \div -12\), \(-48 \div -24\), and \(-96 \div -48\).

Step 3: Calculate Each Ratio

Compute each of the ratios:1. \(-12 \div -6 = 2\)2. \(-24 \div -12 = 2\)3. \(-48 \div -24 = 2\)4. \(-96 \div -48 = 2\)

Step 4: Determine Consistency of Ratios

Since each ratio calculated in Step 3 is 2, the sequence is geometric. All consecutive ratios are equal, confirming the sequence's geometric nature.

Step 5: Identify the Common Ratio

The common ratio for the given sequence is the constant value calculated in the previous steps, which is 2.

Key Concepts

Common Ratio

In a geometric sequence, the 'common ratio' is a critical concept. It is the factor by which you multiply one term to obtain the next term in the sequence. This ratio remains the same throughout the sequence.
For instance, in the sequence \[-6, -12, -24, -48, -96, \ldots \], to find the common ratio, you simply divide each term by its preceding term:

• \(-12 \div -6 = 2\)
• \(-24 \div -12 = 2\)
• \(-48 \div -24 = 2\)
• \(-96 \div -48 = 2\)

All results lead to the same number, \(2\). Thus, the common ratio of this sequence is \(2\). This constant multiplier is what classifies the sequence as geometric.

Consecutive Terms

Consecutive terms in a sequence refer to terms that follow one after another. In the context of a geometric sequence, these terms are connected by multiplication through the common ratio.
When looking at the sequence \[-6, -12, -24, -48, -96, \ldots \], observe the consecutive pairs:

• First and second term: \(-6, -12\)
• Second and third term: \(-12, -24\)
• Third and fourth term: \(-24, -48\)
• Fourth and fifth term: \(-48, -96\)

Each pair of consecutive terms, when divided, shows the common ratio of \(2\). Therefore, the sequence progresses through repeated multiplication by this constant. Understanding these relationships helps you see how the sequence naturally unfolds.

Sequence Properties

Geometric sequences have unique properties that differentiate them from other types of sequences. The most prominent property is the presence of a common ratio, which we've seen is constant across all consecutive terms.
This suggests that if you have a geometric sequence starting with a first term \(a\), each subsequent term can be represented as:\[a, ar, ar^2, ar^3, \ldots \]where \(r\) is the common ratio. Other properties include:

• Exponential growth or decay, depending on whether \(|r| > 1\) or \(|r| < 1\).
• A consistent pattern that simplifies predicting future terms or finding specific terms by using the formula \(a_n = ar^{n-1}\).

In the sequence \[-6,-12,-24,-48,-96, \ldots \], the consistent common ratio of \(2\) shows a doubling pattern, which indicates exponential growth. These properties make geometric sequences easy to understand and apply in various mathematical contexts.