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Chapter 8: Problem 64

Find \(a_{2}\) and \(a_{3}\) for each geometric sequence. $$2, a_{2}, a_{3},-54$$

Short Answer

Expert verified
\(-6, 18\). The second and third terms of the given geometric sequence are -6 and 18 respectively.

Step by step solution

01

Finding the common ratio

Since this is a geometric sequence, the ratio between all consecutive terms should be equal. Let's denote the common ratio as \(r\). Since we know the first and the last terms of the part of sequence provided, and knowing that the fourth term -54 is obtained by multiplying the previous term by the common ratio twice, we can formulate the following equation: \(2 \times r^{3} = -54\) So, to find \(r\) we solve the equation: \(r^{3} = -54/2 = -27\), Therefore, \(r = \sqrt[3]{-27} = -3\).
02

Find \(a_{2}\) and \(a_{3}\)

We found the common ratio \(r = -3\). Since a geometric sequence is defined by \(a_{n} = a_{n-1} \times r\) , we can use this relation to find the missing terms. \(a_{2} = a_{1} \times r = 2 \times -3 = -6\). \(a_{3} = a_{2} \times r = -6 \times -3 = 18\).
03

Check the solution

Once we've calculated the values of \(a_{2}\) and \(a_{3}\), it's always a good idea to check if these values are correct. Doing so we have: \[2, -6, 18, -54\] is indeed a geometric sequence with common ratio of -3.

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