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

Explanation

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

### Step 1: Understanding the Arithmetic Progression
We have the first term \( a = 1 \) and the last term \( l = 31 \). The number of terms including the means is \( m + 2 \) (1 and 31 included).

**Hint:** Remember that the total number of terms in an arithmetic progression is the number of means plus the two endpoints.

### Step 2: Finding the Common Difference
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