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Chapter 5: Problem 64

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

Short Answer

Expert verified
The first six terms of the geometric series are: \(\frac{1}{2}, 1, 2, 4, 8, 16\)

Step by step solution

01

Identify the first term and the common ratio

From the exercise, the first term (\(a_{1}\)) is \(\frac{1}{2}\) and the common ratio (\(r\)) is 2.
02

Calculate the second term (\(a_{2}\))

Use the formula for generating terms in a geometric sequence : \(a_{n} = a_{1} \cdot r^{(n-1)}\), substituting n = 2, \(a_{1}\) = \(\frac{1}{2}\) and r = 2 we get: \(a_{2} = \frac{1}{2} \cdot 2^{(2-1)} = 1\)
03

Calculate the third term (\(a_{3}\))

Using the formula again and substituting n = 3, \(a_{1}\) = \(\frac{1}{2}\) and r = 2, we get: \(a_{3} = \frac{1}{2} \cdot 2^{(3-1)} = 2\)
04

Calculate the fourth term (\(a_{4}\))

Substitute n = 4 in the formula, \(a_{1}\) = \(\frac{1}{2}\) and r = 2, we get: \(a_{4} = \frac{1}{2} \cdot 2^{(4-1)} = 4\)
05

Calculate the fifth term (\(a_{5}\))

Substitute n = 5 in the formula, \(a_{1}\) = \(\frac{1}{2}\) and r = 2, we get: \(a_{5} = \frac{1}{2} \cdot 2^{(5-1)} = 8\)
06

Calculate the sixth term (\(a_{6}\))

Substitute n = 6 in the formula,\(a_{1}\) = \(\frac{1}{2}\) and r = 2, we get: \(a_{6} = \frac{1}{2} \cdot 2^{(6-1)} = 16\)

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Most popular questions from this chapter

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(6,3, \frac{3}{2}, \frac{3}{4}, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{20}\), when \(a_{1}=2, r=2\).

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-8, r=-5\)

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

If you are given a sequence that is arithmetic or geometric, how can you determine which type of sequence it is?
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