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Chapter 9: Problem 2

How is the common ratio of a geometric sequence found?

Short Answer

Expert verified
Divide a consecutive term by its previous term.

Step by step solution

01

Identify Consecutive Terms

In a geometric sequence, pick any two consecutive terms. For example, if the sequence is 2, 4, 8, 16,..., choose two terms such as 4 and 8 or 8 and 16.
02

Divide Second Term by First

To find the common ratio, divide the second term by the first term. Using our example with terms 4 and 8, compute \( r = \frac{8}{4} = 2 \).
03

Check Consistency Across Other Terms

Ensure the quotient is consistent between other consecutive terms to confirm that the sequence is geometric. For example, using terms 8 and 16, compute \( \frac{16}{8} = 2 \). If all ratios are consistent, the common ratio is confirmed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
In a geometric sequence, the common ratio is a fundamental concept. It's the factor that you multiply by to get from one term in the sequence to the next. For instance, in the sequence 2, 4, 8, 16, and so on, the common ratio is 2. This means that each term is obtained by multiplying the previous term by 2.
To find the common ratio, you need to do simple division. You take any two consecutive terms of the sequence and divide the second one by the first. This will give you the common ratio, making it easy to confirm if a sequence is truly geometric.
Consecutive Terms
In the study of sequences, consecutive terms are simply terms that follow one another in the sequence. For a geometric sequence, understanding the consecutive terms is essential because they help in calculating the common ratio.
When dealing with a sequence like 2, 4, 8, 16, identify any two consecutive terms, such as 4 and 8 or 8 and 16. Using these, you can determine the common ratio by performing the division method. Picking these terms carefully ensures that every step, starting from identifying the pattern to verifying it, is clear and straightforward.
Consistency Check
Once you've calculated the common ratio using two consecutive terms, it's crucial to perform a consistency check. This checks if the sequence maintains the same common ratio throughout. Ensuring consistency verifies the integrity of the sequence.
Take additional sets of consecutive terms in the sequence, like 8 and 16, and divide them as before. If the resulting value remains the same, it's a reliable indicator the sequence is geometric and maintains its common ratio. This step helps validate both the pattern and the correctness of your initial division.
Division Method
The division method is a key strategy in identifying a geometric sequence's common ratio. It is straightforward: you divide one term by the preceding term to find the ratio. This simple operation forms the backbone of understanding geometric sequences.
For instance, consider terms like 4 and 8. By dividing 8 by 4, you get 2, which is the common ratio. This method not only helps in finding the common factor but also in confirming whether the sequence's pattern holds consistently across all term pairs. This makes the division method both a fundamental and reliable tool in sequence analysis.

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Most popular questions from this chapter

For the following exercises, find the number of terms in the given finite geometric sequence. \(a_{n}=\\{-1,3,-9, \ldots, 2187\\}\)

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