There are n AM's between 3 and 54.Such that the 8th mean and `(n-2)`th mean is 3 ratio 5. Find n.

Explanation

To solve the problem, we need to find the value of \( n \) such that there are \( n \) arithmetic means (AMs) between 3 and 54, with the condition that the 8th mean and the \( (n-2) \)th mean are in the ratio of 3:5.

### Step-by-Step Solution:

1. **Identify the Arithmetic Sequence**:
Let the arithmetic means be \( a_1, a_2, a_3, \ldots, a_n \). The first term \( a_0 \) is 3, and the last term \( a_{n+1} \) is 54.

2. **Calculate the Common Difference**:
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