\( 20(k-4) ;(k+1) ; m ; 5 k \) is a set of numbers. The first three terms form an arithmetic sequence and the last three form a geometric sequence. Determine the values of \( k \) and \( m \), if \( k, m>0 \).

To solve the problem, we start with the conditions given. The first three terms form an arithmetic sequence, so we set up the equation for these terms: \[ (k - 4) + (k + 1) = 2 \cdot 20. \] Simplifying this, we have: \[ 2k - 3 = 40 \implies 2k = 43 \implies k = 21.5. \] Next, we substitute \( k = 21.5 \) into the three terms: \[ 20(k - 4) = 20(21.5 - 4) = 20 \cdot 17.5 = 350, \] \[ k + 1 = 21.5 + 1 = 22.5. \] Now we know the first three terms: \( 350, 22.5, m \). Next, we examine the last three terms, forming a geometric sequence: \[ (k + 1), m, 5k. \] We will use the property of the geometric sequence, where the square of the middle term equals the product of the other two: \[ m^2 = (k + 1)(5k). \] Substituting \( k = 21.5 \): \[ m^2 = (21.5 + 1)(5 \cdot 21.5) = 22.5 \cdot 107.5. \] Calculating \( 22.5 \cdot 107.5 \): \[ 22.5 \cdot 107.5 = 2428.75. \] Thus: \[ m^2 = 2428.75 \implies m = \sqrt{2428.75} \approx 49.28. \] We have determined: \[ k = 21.5, \quad m \approx 49.28. \] Thus, the values of \( k \) and \( m \) are approximately \( k = 21.5 \) and \( m \approx 49.28 \).