Problem 9 - College Algebra with Corequisite Support

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, it is geometric with a common ratio of 2.

Step 1: Identify the Sequence

Examine the given sequence: -6, -12, -24, -48, -96, ... The task is to determine if this sequence is geometric.

Step 2: Calculate the Ratio of Successive Terms

To check if the sequence is geometric, calculate the ratio between successive terms:\[r = \frac{a_{n+1}}{a_n}\]Calculate the ratio for the first two terms:-12 / -6 = 2.

Step 3: Check Consistency of the Ratio

Continue calculating the ratio for the rest of the pairs:- \(-12, -24\): \(-24 / -12 = 2\)- \(-24, -48\): \(-48 / -24 = 2\)- \(-48, -96\): \(-96 / -48 = 2\) Since the ratio is consistently 2, the sequence is geometric.

Step 4: Conclusion - Determine the Common Ratio

Because the ratio between each pair of successive terms is constant, the sequence is geometric. The common ratio is 2.

Key Concepts

Common Ratio

A geometric sequence is characterized by a constant factor known as the \(\textit{common ratio}\) (\( r \)). This ratio is what each term is multiplied by to get the next term in the sequence. It's a key component that defines the nature of such sequences. For example, consider the sequence \(-6, -12, -24, -48, -96, \ldots\). To find the common ratio, we divide any term by its preceding term.

• Calculate the ratio for the first two terms: \(-12 / -6 = 2\)
• Test other pairs as well to confirm the consistency, such as \(-24 / -12 = 2\)

By ensuring that this ratio stays constant across each pair of terms, we establish that the sequence is indeed geometric, with a common ratio of 2.

Successive Terms

In the context of sequences, \textbf{successive terms} refer to consecutive numbers or values in a series. For geometric sequences, examining these helps identify whether a sequence is truly geometric. Let's break down what we mean when we talk about looking at successive terms:

• Take two terms in a sequence, such as \(-12\) and \(-24\)
• Calculate their ratio: here, \(-24 / -12 = 2\)

When evaluating successive terms in the geometric sequence \(-6, -12, -24, -48, -96, \ldots\), it is crucial they maintain the same ratio. This consistent relationship is key to qualifying the sequence as geometric. Without it, the sequence could be anything else but geometric. Check as many pairs as needed to be sure!

Sequence Consistency

\textbf{Sequence consistency} is an essential trait of geometric sequences. It ensures that the common ratio stays the same no matter which pair of terms you examine. Consistency is the backbone of a well-defined geometric sequence. Here's how to check for it:

• Start by looking at the first two terms and find their ratio, e.g., \(-12 / -6 = 2\).
• Proceed to the next pair, like \(-24 / -12 = 2\), and repeat for others.

This demonstrates that the ratio doesn't change, ensuring the sequence behaves predictably. If a sequence loses this consistency, it cannot be considered geometric. Hence, always keep an eye on this consistency to validate such sequences effectively.
In our sequence \(-6, -12, -24, -48, -96, \ldots\), we've confirmed this trait as the ratio remains constant at 2 throughout.