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Chapter 5: Problem 66

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{5}, r=\frac{1}{2}\)

Short Answer

Expert verified
The first six terms of the geometric sequence are \( \frac{1}{5}, \frac{1}{10}, \frac{1}{20}, \frac{1}{40}, \frac{1}{80}, \frac{1}{160} \)

Step by step solution

01

Understand the geometric sequence

In a geometric sequence, each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number, called the common ratio. In other words, the \(n^{th}\) term \(a_{n}\) of a geometric sequence can be calculated using the formula \(a_{n} = a_{1} * r^{(n-1)}\), where \(r\) is the common ratio, \(a_{1}\) is the first term, and \(n\) is the term number.
02

Calculation of the terms

Given \(a_{1} = \frac{1}{5}\) and \(r = \frac{1}{2}\), the first six terms of the sequence can be calculated as follows: \n \(a_{1} = \frac{1}{5}\), \n \(a_{2} = a_{1}*r = \frac{1}{5} * \frac{1}{2} = \frac{1}{10}\), \n \(a_{3} = a_{2}*r = \frac{1}{10} * \frac{1}{2} = \frac{1}{20}\), \n \(a_{4} = a_{3}*r = \frac{1}{20} * \frac{1}{2} = \frac{1}{40}\), \n \(a_{5} = a_{4}*r = \frac{1}{40} * \frac{1}{2} = \frac{1}{80}\), \n \(a_{6} = a_{5}*r = \frac{1}{80} * \frac{1}{2} = \frac{1}{160}\)
03

Summary of all terms

Then, the first six terms of the given geometric sequence are: \(\frac{1}{5}, \frac{1}{10}, \frac{1}{20}, \frac{1}{40}, \frac{1}{80}, \frac{1}{160}\)

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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, \(r\). \(a_{1}=5000, r=1\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A sequence that is not arithmetic must be geometric.

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\sqrt{3}, 3,3 \sqrt{3}, 9, \ldots\)

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