Introduction:
In this problem, we are given that A1, A2, and A3 are in arithmetic progression (AP). We need to determine the values of p, q, and r such that Ap, Aq, and Ar are also in AP.

Given:
A1, A2, and A3 are in AP.

Definition of Arithmetic Progression (AP):
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is denoted by 'd'.

Deriving the Common Difference:
To find the common difference 'd' in the given sequence A1, A2, A3, we can use the formula:

d = A2 - A1 (as A1 and A2 are consecutive terms in the AP)

Deriving the Terms Ap, Aq, and Ar:
To determine the values of p, q, and r, we need to express Ap, Aq, and Ar in terms of A1, A2, and A3.

Ap can be expressed as A1 + (p-1)d, where 'p' is the position of Ap in the AP.
Aq can be expressed as A1 + (q-1)d, where 'q' is the position of Aq in the AP.
Ar can be expressed as A1 + (r-1)d, where 'r' is the position of Ar in the AP.

Condition for Ap, Aq, and Ar to be in AP:
For Ap, Aq, and Ar to be in AP, the common difference between any two consecutive terms should be the same. Therefore, we need to determine the values of p, q, and r such that the difference between consecutive terms remains constant.

Using the formula for the common difference, we can write the following equations:

d = A2 - A1
d = Aq - Ap
d = Ar - Aq

Solving the Equations:
We can solve the above equations to find the values of p, q, and r.

From the equation d = Aq - Ap, we can substitute the expressions for Ap and Aq:

d = (A1 + (q-1)d) - (A1 + (p-1)d)
Simplifying, we get:
d = qd - pd
d = (q - p)d
1 = q - p

Similarly, from the equation d = Ar - Aq, we can substitute the expressions for Aq and Ar:

d = (A1 + (r-1)d) - (A1 + (q-1)d)
Simplifying, we get:
d = rd - qd
d = (r - q)d
1 = r - q

From the above equations, we can conclude that p, q, and r are consecutive integers.

Conclusion:
If A1, A2, and A3 are in arithmetic progression (AP), then Ap, Aq, and Ar are also in AP if p, q, and r are consecutive integers.