Final answer:
The value of x in the geometric sequence 27, x, 3, 1 is found by equating the ratios between the consecutive terms, solving the resulting equation x^2 = 81, and determining that x = 9 since the sequence is decreasing.
Step-by-step explanation:
To find the value of x in the geometric sequence 27, x, 3, 1, we can use the property that the ratio between consecutive terms is constant. Given that the second term is x and the third term is 3, we can write:
r = x / 27 (ratio from the first to the second term)
r = 3 / x (ratio from the second to the third term)
By equating the two expressions for r, we get:
x / 27 = 3 / x
Cross multiply to get:
x2 = 27 * 3
x2 = 81
Thus, we find:
x = √81
x = 9 or x = -9
However, since x must form a decreasing sequence from 27 down to 1, the negative value does not fit the context. Therefore, the value of x is 9.