Final answer:To find the fifth term of a geometric sequence with a second term of 12 and a fourth term of 108, we first calculate the common ratio. The common ratio is 3, and thus the fifth term is found to be 324.Explanation:To determine the fifth term of a geometric sequence when given the second and fourth terms, we must first find the common ratio. In a geometric sequence, each term after the first is multiplied by a constant called the common ratio to get the next term. We are given that the second term is 12 and the fourth term is 108.Let's denote the first term of the sequence asa, and the common ratio asr. This means the second term isarand the fourth term isar^3. With the given information we have:ar = 12ar^3 = 108By dividing the second equation by the first, we can find the common ratio:\(\frac{ar^3}{ar} = \frac{108}{12} => r^2 = 9\)So the common ratioris 3 because 3^2 = 9. Now we can determine the fifth term, which is ar^4. We already know that ar = 12, so:ar^4 = 12 × 3^3ar^4 = 12 × 27 = 324Thus, the fifth term of the geometric sequence is 324....

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