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 number of arithmetic means (n) between the numbers 3 and 54, given that the 8th mean and the (n-2)th mean are in the ratio of 3:5.

### Step-by-Step Solution:

1. **Understanding the Arithmetic Means**:
- The arithmetic means between 3 and 54 can be represented as an arithmetic progression (AP) with the first term \( a = 3 \) and the last term \( l = 54 \).
- The total number of terms in this AP will be \( n + 2 \) (including the two endpoints).

...