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