Arithmetic Progression (AP) and Geometric Progression (GP) Relationship:
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. If p, q, and r are in AP, then the common difference is d = q - p = r - q.
A Geometric Progression (GP) is a sequence of numbers in which the ratio of any two consecutive terms is constant. If x, y, and z are in GP, then the common ratio is r = y / x = z / y.

Relationship between AP and GP:
When numbers are in both AP and GP, there is a unique relationship between them. Let's assume a, ar, and ar^2 are the terms that are in both AP and GP.
Since a, ar, and ar^2 are in AP:
ar - a = ar^2 - ar
=> r - 1 = r^2 - r
=> r^2 - 2r + 1 = 0
=> (r - 1)^2 = 0
=> r = 1
Therefore, when numbers are in both AP and GP, the common ratio of the GP is always 1. This is a special case where the terms form a constant sequence.

Conclusion:
In conclusion, when p, q, and r are in AP and x, y, and z are in GP, if a, ar, and ar^2 are in both AP and GP, then the common ratio of the GP is always 1. This relationship is a unique property that arises when numbers are in both AP and GP simultaneously.