The three terms x+12, 3x+5 and 2x+25 are the first three terms of an arithmetic progression. What is the value of x?

To find the value of x in this arithmetic progression, we need to find the common difference between consecutive terms.

The common difference (d) in an arithmetic progression is the constant value added (or subtracted) to each term to get the next term.

Given the three terms: x + 12, 3x + 5, and 2x + 25.

To find the common difference between the first and second term:

Common difference (d) = (Second term) - (First term)
d = (3x + 5) - (x + 12)
d = 3x + 5 - x - 12
d = 2x - 7

To find the common difference between the second and third term:

Common difference (d) = (Third term) - (Second term)
d = (2x + 25) - (3x + 5)
d = 2x + 25 - 3x - 5
d = -x + 20

Since the common difference in an arithmetic progression should be the same for any pair of consecutive terms, we can set the expressions for the two common differences equal to each other:

2x - 7 = -x + 20

Now, we can solve this equation to find the value of x:

2x + x = 20 + 7
3x = 27
x = 27/3
x = 9

Therefore, the value of x is 9.