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Chapter 5: Problem 95

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. \(1.5,-3,6,-12, \ldots\)

Short Answer

Expert verified
The seventh term of the sequence is -192.

Step by step solution

01

Identify the Common Ratio

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. To find the common ratio, divide the second term by the first term, which is \(-3/1.5 = -2\). Therefore, the common ratio (r) for the sequence is -2.
02

Derive the General Term

The general (nth) term of a geometric sequence can be represented by the formula \(a_n = a_1 * r^{(n-1)}\), where \(a_n\) is the nth term, \(a_1\) is the first term, and r is the common ratio. Thus, for our sequence, the formula for the general term becomes \(a_n = 1.5 * (-2)^{(n-1)}\).
03

Calculate the Seventh Term

To find the seventh term (\(a_{7}\)), substitute n = 7 into the general formula. Thus, \(a_{7} = 1.5 * (-2)^{(7-1)} = -192\).

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{4}, r=2\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{6}\), when \(a_{1}=-2, r=-3\).

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

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(7,-7,-21,-35, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=1000, r=1\)
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