Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A. P. and the ratio of `7^(t h)`and `(m-1)^(t h)`numbers is 5 : 9. Find the value of m.

Explanation

To solve the problem, we need to find the value of \( m \) given that the numbers inserted between 1 and 31 form an arithmetic progression (A.P.) and the ratio of the 7th term to the \( (m-1) \)th term is \( 5:9 \).

### Step-by-Step Solution:

1. **Identify the Terms of the A.P.**:
- Let the first term of the A.P. be \( a \) and the common difference be \( d \).
- The terms of the A.P. can be expressed as:
- \( A_1 = a \)
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