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 will find the value of \( m \) given that \( m \) arithmetic means are inserted between 1 and 31, and the ratio of the 7th mean to the \( (m-1) \)th mean is \( 5:9 \).

### Step 1: Set up the arithmetic progression
Let the arithmetic means be \( a_1, a_2, a_3, \ldots, a_m \). The complete sequence will be:
\[ 1, a_1, a_2, \ldots, a_m, 31 \]

### Step 2: Determine the common difference
The total number of terms in the sequence is \( m + 2 \) (including 1 and 31). The first term \( a = 1 \) and the last term \( l = 31 \).
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