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Chapter 5: Problem 97

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. \(0.0004,-0.004,0.04,-0.4, \ldots\)

Short Answer

Expert verified
-40

Step by step solution

01

Identify the first term and the common ratio

For the given geometric sequence \(0.0004,-0.004,0.04,-0.4, \ldots\), the first term \(a_{1}\) can be seen to be 0.0004. The common ratio \(r\) can be found by dividing the second term by the first term (or any term by its preceding term), which gives \(r = -0.004 / 0.0004 = -10\).
02

Formulate the general term (the nth term)

Plug in the values of \(a_{1}\) and \(r\) into the formula for the nth term of a geometric sequence, which is \(a_{n} = a_{1} * r^{(n-1)}\). Hence, the general term \(a_{n}\) is given by \(a_{n} = 0.0004 * (-10)^{(n-1)}\).
03

Find the 7th term

To find the 7th term \(a_{7}\), simply replace \(n\) with 7 in the general term: \(a_{7} = 0.0004 * (-10)^{(7-1)} = 0.0004 * (-10)^6 = -40\). Note that the terms of the sequence alternate in sign, which matches the negative sign in the 7th term.

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. \(a_{1}=-20, d=-4\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{4}\), when \(a_{1}=9, r=-\frac{1}{3}\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{8}\), when \(a_{1}=1,000,000, r=0.1\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{7}\), when \(a_{1}=4, r=2\)

What is the common difference in an arithmetic sequence?
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