If -8, m, n, 19 are in arithmetic progression, find (m, n)

To solve this problem, we need to use the arithmetic progression formula: a_n = a_1 + (n-1)d where a_n is the nth term of the arithmetic progression, a_1 is the first term, n is the number of terms, and d is the common difference. We are given that -8, m, n, and 19 are in arithmetic progression. So we can set up the following equations: m = -8 + d n = -8 + 2d 19 = -8 + 3d We can solve for d by subtracting the first equation from the second equation: n - m = 2d - d n - m = d We can substitute this expression for d into the third equation: 19 = -8 + 3(n - m) Simplifying this equation gives: 27 = 3(n - m) 9 = n - m We can substitute this expression for n - m into the equation we derived earlier: n - m = d So we have: d = 9 Substituting this value of d into any of the earlier equations will allow us to solve for m and n. For example, using the equation: m = -8 + d gives: m = -8 + 9 m = 1 And using the equation: n = -8 + 2d gives: n = -8 + 2(9) n = 10 Therefore, the solution is (m, n) = (1, 10)