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Chapter 8: Problem 19

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. $$18,6,2, \frac{2}{3}, \dots$$

Short Answer

Expert verified
The general term of the sequence is given by \(a_{n} = 18 * (\frac{1}{3})^{(n-1)}\). The seventh term of this sequence is \(\frac{2}{243}\).

Step by step solution

01

Calculate the common ratio

Given the sequence \(18, 6, 2, \frac{2}{3}, \ldots\) the ratio between each term is calculated as the second term divided by the first term, or \(r = \frac{b}{a}\). Hence the common ratio \(r\) is \(\frac{6}{18} = \frac{1}{3}\).
02

Derive the general formula

The general formula for a geometric sequence is given by \(a_{n} = a_{1} * r^{(n-1)}\), where \(a_{1}\) is the first term and \(r\) is the common ratio. Plugging our values in, we get \(a_{n} = 18 * (\frac{1}{3})^{(n-1)}\)
03

Find the seventh term

To find the seventh term of a geometric sequence, all that has to be done is to replace \(n\) with 7 in our derived formula. Thus, \(a_{7} = 18 * (\frac{1}{3})^{(7-1)} = 18 * (\frac{1}{3})^{6} = \frac{2}{243}\).

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Most popular questions from this chapter

Exercises \(85-87\) will help you prepare for the material covered in the next section. Use the formula \(a_{n}-a_{1} 3^{n-1}\) to find the seventh term of the sequence \(11,33,99,297, \dots\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It makes a difference whether or not I use parentheses around the expression following the summation symbol, because the value of \(\sum_{i=1}^{5}(i+7)\) is \(92,\) but the value of \(\sum_{i=1}^{8} i+7\) is $43 .

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{n !}{(n-1) !}-\frac{1}{n-1}$$

Exercises will help you prepare for the material covered in the next section. In Exercises \(112-113,\) show that $$ 1+2+3+\cdots+n-\frac{n(n+1)}{2} $$is true for the given value of \(n .\) $$n-5 \text { : Show that } 1+2+3+4+5-\frac{5(5+1)}{2}.$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I modeled California's population growth with a geometric sequence, so my model is an exponential function whose domain is the set of natural numbers.
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