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Chapter 5: Problem 53

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

Short Answer

Expert verified
The first six terms of the geometric sequence are: \(3, -6, 12, -24, 48, -96\).

Step by step solution

01

- Understand the Formula

The formula for the \(n^{th}\) term, \(a_{n}\), of a geometric sequence is given as: \(a_{n} = a_{1} * r^{(n-1)}\), where \(a_{1}\) is the initial term, \(r\) is the common ratio, and \(n\) is the position of the term.
02

- Find the First Term (\(a_{1}\))

The first term of the sequence is provided in the problem. We take \(a_{1}\) = 3 as the first term.
03

- Find the Second Term (\(a_{2}\))

By substituting \(n=2, a_{1}=3, r=-2\) into the formula, we can calculate the second term. \(a_{2} = a_{1} * r^{(2-1)} = 3 * (-2)^{1} = -6\).
04

- Find the Third Term (\(a_{3}\))

Next, calculate the third term, \(a_{3}\), by substituting \(n=3, a_{1}=3, r=-2\) into the formula, we get \(a_{3} = 3 * (-2)^{2} = 12\).
05

- Find the Fourth, Fifth, and Sixth Terms

Continue in the same pattern up to the sixth term: \(a_{4} = 3 * (-2)^{3} = -24, a_{5} = 3 * (-2)^{4} = 48, a_{6} = 3 * (-2)^{5} = -96\).

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