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The concept of a "common ratio" is crucial for understanding geometric sequences. When a sequence is geometric, the ratio of any term to the previous term is consistent across the entire sequence. This consistent value is known as the common ratio, denoted by \( r \).

This common ratio helps us understand how the sequence progresses from one term to the next. For example, in our sequence \( 9, -6, 4, -\frac{8}{3}, \ldots \), every term beyond the first can be calculated by multiplying the previous term by \(-2/3\).

Recognizing the common ratio is critical because it reveals the pattern that defines the sequence. In exercises, you will often need to find this ratio to determine if a given sequence is geometric.

"Sequence terms" are the individual elements or numbers that make up a sequence. In our exercise, the sequence terms are \( a_1 = 9 \), \( a_2 = -6 \), \( a_3 = 4 \), and \( a_4 = -\frac{8}{3} \).

Understanding sequence terms helps us to perform operations like finding differences or calculating ratios. When identifying sequence terms, it is helpful to label them clearly.

• \( a_1 = 9 \)
• \( a_2 = -6 \)
• \( a_3 = 4 \)
• \( a_4 = -\frac{8}{3} \)

By labeling these terms, you can break down the sequence into manageable parts for further analysis, like calculating the common ratio.

To "identify a sequence" means to determine the type of pattern it follows. In mathematics, there are various types of sequences such as arithmetic, geometric, and others. In our task, we needed to determine if the sequence \( 9, -6, 4, -\frac{8}{3}, \cdots \) is geometric.

To identify whether a sequence is geometric, you'll look at the ratios of consecutive terms. If the ratios between consecutive terms are all the same, then it's a geometric sequence.

Recognizing and identifying the type of sequence is a foundational step before solving further problems related to that sequence, like finding missing terms or calculating the sum.

"Comparing ratios" is a straightforward yet vital process when dealing with sequences. Once you've identified the sequence terms, you compare the ratios of consecutive terms to see if they're constant. This constant ratio confirms a geometric sequence.

In our sequence, we calculated the ratios as follows:
\( \frac{-6}{9} = -\frac{2}{3} \)
\( \frac{4}{-6} = -\frac{2}{3} \)
\( \frac{-\frac{8}{3}}{4} = -\frac{2}{3} \)

This demonstrated that each term divided by its preceding term always equals \(-\frac{2}{3}\). By comparing these ratios, we confirmed that the sequence is geometric, verifying that the constant value, \(-\frac{2}{3}\), is indeed the common ratio.