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Chapter 5: Problem 92

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. \(3,15,75,375, \ldots\)

Short Answer

Expert verified
The formula for the nth term of the sequence is \(a_{n} = 3 \cdot 5^{(n-1)}\), and the seventh term of the sequence is 46875.

Step by step solution

01

Identify the first term and the common ratio

For the given sequence \(3, 15, 75, 375\), the first term \(a = 3\). The ratio \(r\) can be found by dividing the second term by the first term. So, \(r = 15 / 3 = 5\). Thus, the first term 'a' is 3 and the common ratio 'r' is 5.
02

Formulate the nth term formula

Using the formula of a geometric sequence, \(a_{n} = a \cdot r^{(n-1)}\), which becomes \(a_{n} = 3 \cdot 5^{(n-1)}\) with our identified values.
03

Find the 7th term of the sequence

Substituting \(n = 7\) into our formula \(a_{7} = 3 \cdot 5^{(7-1)}\), we find that \(a_{7} = 3 \cdot 5^{6}\). Computing this, we find that the seventh term \(a_{7} = 46875\).

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Most popular questions from this chapter

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.

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

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 \(a_{8}\), when \(a_{1}=12, r=\frac{1}{2}\).

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.
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