Problem 64 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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, \(r = \sqrt[3]{-27} = -3\).

Step 2: 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. \(a_{2} = a_{1} \times r = 2 \times -3 = -6\). \(a_{3} = a_{2} \times r = -6 \times -3 = 18\).

Step 3: 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.