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Chapter 5: Problem 58

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=3000, r=-1\)

Short Answer

Expert verified
The first six terms of the geometric sequence are: \(3000\), \(-3000\), \(3000\), \(-3000\), \(3000\), and \(-3000\).

Step by step solution

01

Identification

First, identify the first term and the common ratio from the problem. From the problem statement, it can be seen that the first term \(a_{1} = 3000\) and the common ratio \(r = -1\).
02

Apply Geometric Sequence Formula

Use the formula for finding the \(n^{th}\) term of a geometric sequence which is \(a_n = a_1 \cdot r^{(n-1)}\). For finding the first six terms, you need to replace \(n\) with the corresponding term number.
03

Find the First Six Terms

Apply the formula six times, for \(n = 1\) to \(n = 6\). For the first term, the exponent on the common ratio will be \(0\) (since \(n = 1\) and thus \(1 - 1 = 0\)). Any number raised to the power \(0\) equals \(1\), so the first term equals the initial term, which is \(3000\). For the second term, \(n = 2\) thus the exponent is \(1\), and the term equals \(3000 times -1\). Continue in this way to find the subsequent terms.
04

Construct the Sequence

Finally, construct the geometric sequence with the calculated terms. Remember the sign of the terms will alternate between positive and negative due to the negative common ratio.

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Most popular questions from this chapter

Use the appropriate formula shown above to find \(2+4+6+8+\cdots+200\), the sum of the first 100 positive even integers.

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=3\)

A professional baseball player signs a contract with a beginning salary of $$\$ 3,000,000$$ for the first year with an annual increase of \(4 \%\) per year beginning in the second year. That is, beginning in year 2 , the athlete's salary will be \(1.04\) times what it was in the previous year. What is the athlete's salary for year 7 of the contract? Round to the nearest dollar.

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=-3\)

What is the common ratio in a geometric sequence?
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