If the sum of the first n terms of the sequence is 76.125 then the value of n should be 4.
To find the number of terms (n) in the geometric sequence where the first term is 6, the common ratio is 1.5, and the sum of the first n terms is 76.125, we can use the formula for the sum of a finite geometric series:
Sn = a(1 - r^n) / (1 - r)
where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Plugging in the given values:
76.125 = 6(1 - 1.5^n) / (1 - 1.5)
Solving for n:
n = (log(13.75) - log(6)) / log(1.5)
n ≈ 3.48
Since the number of terms has to be a whole number, we round up to the nearest integer.
Therefore, n = 4.