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Chapter 5: Problem 59

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

Short Answer

Expert verified
The first six terms of the geometric sequence with \(a_{1} = -2\) and \(r = -3\) are -2, 6, -18, 54, -162, 486

Step by step solution

01

Understanding the formula

The formula to find the n-th term of a geometric sequence is \(a_{n} = a_{1} \cdot r^{(n-1)}\), where \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. In this situation, \(a_{1} = -2\) and \(r = -3\).
02

finding the first term

The first term is simply the given starting term \(a_{1}\), which is -2 in this case.
03

finding the second term

By substituting \(n = 2\), \(a_{1} = -2\), and \(r = -3\) into the formula, the second term (\(a_{2}\)) can be calculated as follows: \(a_{2} = a_{1} \cdot r^{(n-1)} = -2 \cdot (-3)^{(2-1)} = -2 \cdot -3 = 6\).
04

finding the third term

Substitute \(n = 3\), \(a_{1} = -2\), and \(r = -3\) into the formula, the third term (\(a_{3}\)) can be calculated as follows: \(a_{3} = a_{1} \cdot r^{(n-1)} = -2 \cdot (-3)^{(3-1)} = -2 \cdot 9 = -18\).
05

finding the remaining terms

Similarly, by substituting into the formula for the \(n = 4, 5, 6\) terms, it's found that: \(a_{4} = a_{1} \cdot r^{(n-1)} = -2 \cdot (-3)^{(4-1)} = 54\); \(a_{5} = a_{1} \cdot r^{(n-1)} = -2 \cdot (-3)^{(5-1)} = -162\); \(a_{6} = a_{1} \cdot r^{(n-1)} = -2 \cdot (-3)^{(6-1)} = 486\).

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Most popular questions from this chapter

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010. Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,24.4 \%\) of American men ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.3\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American men ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American men ages 25 and older who will be college graduates by \(2019 .\)

You are offered a job that pays $$\$ 30,000$$ for the first year with an annual increase of \(5 \%\) per year beginning in the second year. That is, beginning in year 2 , your salary will be \(1.05\) times what it was in the previous year. What can you expect to earn in your sixth year on the job? Round to the nearest dollar.

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

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)

Suppose you save $$\$ 1$$ the first day of a month, $$\$ 2$$ the second day, $$\$ 4$$ the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the thirtieth day of the month?
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