There are m arithmetic means between 5 and -16 such that the ratio of the 7th mean to the (m-7)th mean is 1:4. Find the value of m.

Explanation

To solve the problem, we need to find the value of \( m \) given that there are \( m \) arithmetic means between 5 and -16, and the ratio of the 7th mean to the (m-7)th mean is 1:4.

### Step-by-Step Solution:

1. **Identify the terms in the arithmetic progression (AP)**:
The first term \( a_1 = 5 \) and the last term \( a_{m+2} = -16 \). The total number of terms in the AP is \( m + 2 \).

2. **Use the formula for the nth term of an AP**:
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