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Chapter 5: Problem 94

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. \(12,6,3, \frac{3}{2}, \ldots\)

Short Answer

Expert verified
The formula for the nth term of the geometric sequence is \(a_{n} = 12 \times (0.5)^{n-1}\) and the seventh term of the sequence, \(a_{7}\), is 0.1875.

Step by step solution

01

Identify the First Term and the Common Ratio

A geometric sequence's nth term can be calculated using the formula \(a_{n} = a_{1} \times r^{n-1}\), where \(a_{1}\) is the first term, \(r\) is the common ratio, and n is the term number. In the given sequence, \(a_{1} = 12\) and \(r\) can be calculated by dividing the second term by the first term, which is \(0.5 = 6/12\).
02

Write Down the Formula for the nth Term

The formula for the nth term of the given geometric sequence can be written as \(a_{n} = 12 \times (0.5)^{n-1}\).
03

Find the Seventh Term of the Sequence

Substitute \(n = 7\) into the formula to find the seventh term of the sequence, \(a_{7}\). This gives \(a_{7} = 12 \times (0.5)^{7-1} = 12 \times 0.5^6\).
04

Evaluate the Expression

Calculate the value of the expression to find \(a_{7} = 12 \times 0.5^6 = 0.1875\).

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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, \(r\). \(a_{1}=\frac{1}{4}, r=2\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sequence \(1,4,8,13,19,26, \ldots\) is an arithmetic sequence.

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

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