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Chapter 5: Problem 113

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\sqrt{5}, 5,5 \sqrt{5}, 25, \ldots\)

Short Answer

Expert verified
The sequence \(\sqrt{5}, 5,5 \sqrt{5}, 25, \ldots\) is geometric with a common ratio of \(\sqrt{5}\). The next two terms in the sequence are 25\(\sqrt{5}\) and 125.

Step by step solution

01

Check if it's an arithmetic sequence

To verify if a sequence is arithmetic, subtract each term from the one that follows it. Doing so for the first few terms, we get 5 - \(\sqrt{5}\), 5\(\sqrt{5}\) - 5, and 25 - 5\(\sqrt{5}\). The results are not equal, hence the sequence is not arithmetic.
02

Check if it's a geometric sequence

To determine if a sequence is geometric, divide each term by the preceding term. This gives 5 / \(\sqrt{5}\), 5\(\sqrt{5}\) / 5, and 25 / 5\(\sqrt{5}\). Simplifying these expressions, you realize you end up with \(\sqrt{5}\) each time, hence the sequence is geometric and \(\sqrt{5}\) is the common ratio.
03

Find the next two terms of the sequence

Knowing the sequence is geometric and having identified the common ratio, multiply the last given term by the common ratio to find the next terms. Therefore, the next term will be 25\(\sqrt{5}\), and the term after that would be 125.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-2, r=-3\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{4}\), when \(a_{1}=9, r=-\frac{1}{3}\).

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=-3\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{2}, r=2\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{30}\), when \(a_{1}=2, r=-1\).
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