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Chapter 5: Problem 79

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

Short Answer

Expert verified
The 30th term \(a_{30}\) of the sequence is -2.

Step by step solution

01

Calculating Common Ratio

The common ratio in a geometric sequence is given by the ratio between any term and its previous term. In this exercise the common ratio \(r\) is given as -1.
02

Understanding the Term Expression

The \(n^{th}\) term of a geometric sequence can be calculated using the formula: \(a_{n} = a_{1} \cdot r^{n-1}\). From this expression, \(a_{n}\) is the term index, \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
03

Substitute and Solve

Substitute the first term \(a_{1}\) as 2, common ratio \(r\) as -1 and term number \(n\) as 30 into the expression to find the 30th term of the sequence: \(a_{30} = a_{1} \cdot r^{n-1} = 2 \cdot (-1)^{30-1} = 2 \cdot (-1)^{29}\). Because the exponent of -1 is an odd number, the result becomes -2.

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Most popular questions from this chapter

Company A pays $$\$ 24,000$$ yearly with raises of $$\$ 1600$$ per year. Company B pays $$\$ 28,000$$ yearly with raises of $$\$ 1000$$ per year. Which company will pay more in year 10 ? How much more?

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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. \(0.0007,-0.007,0.07,-0.7, \ldots\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(15,30,60,120, \ldots\)
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