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Chapter 11: Problem 1

Find the common ratio. $$2,4,8,16, \dots$$

Short Answer

Expert verified
The common ratio is 2.

Step by step solution

01

Identify Consecutive Terms

To find the common ratio in a geometric sequence, first identify two consecutive terms. For example, here are the first two terms: 2 and 4.
02

Calculate the Ratio

Divide the second term by the first term. In this case, the ratio is found by dividing 4 by 2: \[ \frac{4}{2} = 2 \]
03

Confirm the Ratio

To verify the common ratio, calculate the ratio between another pair of consecutive terms. For example, the third term (8) divided by the second term (4): \[ \frac{8}{4} = 2 \] Since the ratio is consistent (2), 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 the factor by which we multiply each term to get the next term. This ratio remains consistent throughout the sequence. To find it, you need any two consecutive terms from the sequence.
consecutive terms
Consecutive terms are terms that follow one another in the sequence without any gaps. For example, in the sequence 2, 4, 8, 16, the pairs (2, 4), (4, 8), and (8, 16) are consecutive terms. To find the common ratio, identify any two of these pairs.
verify ratio
To verify the common ratio, calculate the ratio between a different set of consecutive terms and check if the result is the same. For instance, dividing the third term (8) by the second term (4) in our example gives \(\frac{8}{4} = 2\). Having consistent results confirms the accuracy of the common ratio found initially.

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