Between 1 and 31 are inserted `m` arithmetic mean so that the ratio of the `(7)^(th)` and `(m-1)^(th)` means is 5:9. Find the value of `mdot`

Explanation

To solve the problem step by step, we will find the value of \( m \) such that the ratio of the 7th and \( (m-1)^{th} \) arithmetic means inserted between 1 and 31 is \( 5:9 \).

### Step 1: Understanding the Problem
We need to insert \( m \) arithmetic means between the numbers 1 and 31. This creates a sequence of \( m + 2 \) terms: \( 1, A_1, A_2, \ldots, A_m, 31 \).

### Step 2: Finding the Common Difference
The common difference \( d \) of the arithmetic progression can be calculated using the formula:
\[
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