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Chapter 5: Problem 105

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \ldots\)

Short Answer

Expert verified
The sequence is geometric with a common ratio of \( \frac{1}{2} \) and the next two terms are \( \frac{3}{16}, \frac{3}{32} \).

Step by step solution

01

Identify the type of sequence

A sequence can be identified as geometric if each term is a constant multiple of the preceding term. That is, each term can be calculated by multiplying the previous term by a constant. This constant is known as the common ratio, \(r\). To set out and test this hypothesis, we can calculate the ratio between sequential terms. For example, the ratio between the first and second term in the sequence is given by \( \frac{3/2}{3} = \frac{1}{2} \) . We then find the same ratio for the third term to the second term \( \frac{3/4}{3/2} = \frac{1}{2} \), and we see this pattern continues, indicating a geometric sequence.
02

Determining the next two terms.

Since we have identified the sequence as geometric with a common ratio of \( \frac{1}{2} \), we can easily determine the next term by multiplying the last term by the common ratio. The last term given is \( \frac{3}{8} \). So, the next term would be \( \frac{3}{8} \cdot \frac{1}{2} = \frac{3}{16} \). Following the same pattern, the term after that would be \( \frac{3}{16} \cdot \frac{1}{2} = \frac{3}{32} \). So, the next two terms in the sequence would be \( \frac{3}{16} \) and \( \frac{3}{32} \).

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Most popular questions from this chapter

A professional baseball player signs a contract with a beginning salary of $$\$ 3,000,000$$ for the first year with an annual increase of \(4 \%\) per year beginning in the second year. That is, beginning in year 2 , the athlete's salary will be \(1.04\) times what it was in the previous year. What is the athlete's salary for year 7 of the contract? Round to the nearest dollar.

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