Final answer:
The expression for the nth term of the given sequence is an = (1/2) * (-3)^(n-1) / 2^(n-1), which accounts for the pattern of multiplying by -3 and doubling the denominator.
Step-by-step explanation:
The student is asking for an expression for the apparent nth term of a given sequence. To find this expression, we need to analyze the sequence and identify a pattern. The given sequence is 1/2, -3/4, 9/8, -27/16, and so on.
At first glance, we can see that the numerator is being multiplied by -3 each time and the denominator is doubling, which suggests a common ratio in a geometric sequence. Moreover, the sequence alternates in sign (+, -, +, -), which can be represented by (-1)n+1 or (-1)n, depending on whether n starts from 1 or 0. However, to write an explicit formula for the sequence, we also need to establish the base case when n = 1.
Considering these observations, the general term of the sequence, an, for n ≥ 1, can be expressed as an = (1/2) * (-3)n-1 / 2n-1. This formula accounts for the initial term, the multiplication by -3 of the numerator for each successive term and the doubling of the denominator, all incorporated into a formulaic representation of n terms in the sequence.