The given arithmetic progression (A.P) has the first three terms as b, c, and 2b. We need to find the ratio of b and c.

Step 1: Understanding Arithmetic Progression (A.P)
Arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant difference is denoted by 'd'. In an A.P, the nth term is given by the formula:
a_n = a_1 + (n - 1)d
where a_n is the nth term, a_1 is the first term, n is the position of the term, and d is the common difference.

Step 2: Identifying the First Three Terms
According to the given information, the first three terms of the A.P are b, c, and 2b. Let's denote them as a_1, a_2, and a_3 respectively.

Step 3: Finding the Common Difference
To find the common difference (d), we can use the formula:
d = a_2 - a_1
Substituting the values, we get:
d = c - b

Step 4: Finding the Ratio of b and c
To find the ratio of b and c, we can use the formula:
ratio = b / c

Step 5: Substituting the Values
To find the ratio, substitute the values of b and c from the given information:
ratio = b / c

Step 6: Simplifying the Ratio
Since we don't have any additional information or constraints, we cannot simplify the ratio any further. Hence, the final answer is:
ratio = b / c

Conclusion
The ratio of b and c in the given arithmetic progression is b / c. The common difference can also be found as c - b.