Answer:
n = 7
Step-by-step explanation:
To find the value of n, we can use the given ratio between the (n-1)th mean and the 4th mean, which is 9:4.
Here's how we can solve the problem step-by-step:
1. Let's start by determining the common ratio between the consecutive terms in the geometric sequence. We can do this by taking the ratio of the 4th mean to the (n-1)th mean:
(n-1)th mean / 4th mean = 9/4
2. We can express the (n-1)th mean and the 4th mean in terms of the initial term (16/27) and the common ratio (r):
(16/27) * r^(n-1) / (16/27) * r^4 = 9/4
3. Simplifying the expression, we can cancel out the common terms:
r^(n-1) / r^4 = 9/4
4. Apply the rule of exponents to the left side of the equation:
r^((n-1)-4) = 9/4
5. Simplify the exponents:
r^(n-5) = 9/4
6. Since the bases are the same, the exponents must be equal:
n - 5 = 2
7. Solve for n:
n = 2 + 5
n = 7
Therefore, the value of n is 7.
