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Chapter 5: Problem 93

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. \(18,6,2, \frac{2}{3}, \ldots\)

Short Answer

Expert verified
The formula for the nth term of the sequence is \(a_{n} = 18 \times \left(\frac{1}{3}\right)^{n-1}\), and the seventh term of the sequence is \(a_{7} = 18 \times \left(\frac{1}{3}\right)^{6}\).

Step by step solution

01

Find the common ratio

To find the common ratio, divide the second term of the sequence by the first: \(r = \frac{6}{18} = \frac{1}{3}\). Therefore, the common ratio \(r\) is \(\frac{1}{3}\).
02

Write the formula for the nth term

Now, using the formula \(a_{n} = a_{1} \times r^{(n-1)}\), let's substitute the values we have: the first term \(a_{1} = 18\) and the common ratio \(r = \frac{1}{3}\). This gives us \(a_{n} = 18 \times \left(\frac{1}{3}\right)^{n-1}\). This is the formula for the nth term of the sequence.
03

Find the seventh term

To find \(a_{7}\), substitute \(n=7\) into the nth term formula: \(a_{7} = 18 \times \left(\frac{1}{3}\right)^{7-1} = 18 \times \left(\frac{1}{3}\right)^{6}\).

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Most popular questions from this chapter

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(-7,-2,3,8, \ldots\)

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of California for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 & 35.60 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project California's population, in millions, for the year 2020 . Round to two decimal places.

Company A pays $$\$ 24,000$$ yearly with raises of $$\$ 1600$$ per year. Company B pays $$\$ 28,000$$ yearly with raises of $$\$ 1000$$ per year. Which company will pay more in year 10 ? How much more?

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

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-8, r=-5\)
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