\( 20(k-4) ;(k+1) ; m ; 5 k \) 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 determine the values of \( k \) and \( m \) from the given sequences, we start with the conditions for arithmetic and geometric sequences. First, since \( 20(k-4) \), \( k+1 \), and \( m \) form an arithmetic sequence, we can set up the equation based on the property of arithmetic sequences: \[ 2(k+1) = 20(k-4) + m. \] Next, since \( k+1 \), \( m \), and \( 5k \) form a geometric sequence, we can use the property that the square of the middle term equals the product of the outer terms: \[ (k+1)(5k) = m^2. \] Now, we can solve these equations step by step. 1. Expanding the first equation: \[ 2(k+1) = 20(k-4) + m, \] gives us: \[ 2k + 2 = 20k - 80 + m, \] which simplifies to: \[ m = 2k + 82 - 20k, \] or \[ m = -18k + 82. \] 2. Next, substitute \( m \) into the second equation: \[ (k+1)(5k) = (-18k + 82)^2. \] Expanding both sides: \[ 5k^2 + 5k = 324k^2 - 2952k + 6724. \] Rearranging gives: \[ 0 = 319k^2 - 2957k + 6724. \] 3. Using the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 319, b = -2957, c = 6724 \): Discriminant: \[ (-2957)^2 - 4 \cdot 319 \cdot 6724. \] Calculate \( \Delta \): \[ 8750449 - 8566088 = 14361. \] Thus, \[ k = \frac{2957 \pm \sqrt{14361}}{2 \times 319} = \frac{2957 \pm 119.8}{638}. \] This results in two solutions: \[ k_1 \approx \frac{3076.8}{638} \approx 4.82, \] \[ k_2 \approx \frac{2837.2}{638} \approx 4.44. \] 4. Choose \( k \approx 4.82 \) (as \( k \) must be positive). Substitute back to find \( m \): \[ m = -18(4.82) + 82 \approx 0.76. \] Final values are \( k \approx 4.82 \) and \( m \approx 0.76 \).