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Chapter 5: Problem 69

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=0.1\)

Short Answer

Expert verified
The first six terms of the geometric sequence are 2, 0.2, 0.02, 0.002, 0.0002, 0.00002

Step by step solution

01

Understand the Variables of a Geometric Sequence

In a geometric sequence, the common ratio (\(r\)) is a constant value we multiply by to get from one term to the next. The first term of the sequence is denoted as \(a_{1}\). Here, it's given that \(a_{1}=2\) and \(r=0.1\)
02

Calculate the Second Term

The second term (\(a_{2}\)) of the sequence is found by multiplying the first term (\(a_{1}\)) by the common ratio (\(r\)). So, using the formula \(a_{2} = a_{1} \cdot r\), we substitute the given \(a_{1}\) and \(r\) into the formula which gives \(a_{2} = 2 \cdot 0.1 = 0.2\)
03

Calculate the Third Term

The third term (\(a_{3}\)) is found by multiplying the second term (\(a_{2}\)) by the common ratio (\(r\)). Substitute the calculated \(a_{2}=0.2\) and given \(r=0.1\) into the formula \(a_{3} = a_{2} \cdot r\) which gives \(a_{3}= 0.2 \cdot 0.1 = 0.02\)
04

Calculate the Fourth to Sixth Terms

Repeat this process of multiplying the last term with \(r=0.1\) for the fourth (\(a_{4}=0.02 \times 0.1 = 0.002\)), fifth (\(a_{5}=0.002 \times 0.1 = 0.0002\)), and sixth terms (\(a_{6}=0.0002 \times 0.1 = 0.00002\))
05

Write Down the Sequence

Finally, write down the first six terms of the sequence as 2, 0.2, 0.02, 0.002, 0.0002, 0.00002

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(18,6,2, \frac{2}{3}, \ldots\)

For the first 30 days of a flu outbreak, the number of students on your campus who become ill is increasing. Which is worse: The number of students with the flu is increasing arithmetically or is increasing geometrically? Explain your answer.

The sum, \(S_{n}\), of the first n terms of an arithmetic sequence is given by $$ S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right), $$ in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by $$ S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}, $$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(7,19,31,43, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{40}\), when \(a_{1}=6, r=-1\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(3,8,13,18, \ldots\)
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