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Chapter 5: Problem 88

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{30}\), when \(a_{1}=8000, r=-\frac{1}{2}\).

Short Answer

Expert verified
The 30th term of the sequence, \(a_{30}\), equals to -\frac{8000}{2^{29}}.

Step by step solution

01

Write down the given

From the problem, the first term \(a_{1}=8000\) and the common ratio \(r= -\frac{1}{2}\) is given and we need to find the 30th term which is \(a_{30}\). The formula to find the nth term of a geometric sequence is \(a_{n} = a_{1}*r^{(n-1)} \).
02

Plug in the given values into the formula

Next, substitute the given values into the formula. Here \(a_{1}=8000\), \(r= -\frac{1}{2}\), \(n=30\). So, the formula becomes \(a_{30} = 8000*(-\frac{1}{2})^{(30-1)}\).
03

Solve the equation

Now, using the rule of exponents to simplify the equation, \(a_{30} = 8000*(-\frac{1}{2})^{29}\). Since -1/2 raised to an odds power results in a negative value, the result is \(a_{30} = -\frac{8000}{2^{29}}\).

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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.

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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. \(3,12,48,192, \ldots\)

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