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Write an expression for the apparent nth term of the sequence. (Assume that n begins with

1/2 , -3/4 , 9/8 , 5/15, −27/16, …
an = ____

User Etan
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1 Answer

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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.

User Shashank Tomar
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