A, B and C are terms of an AP and 2^A , 2^B and 2^C are terms of a GP. Are there any integer examples for A, B and C?

Since A,  B and C are in AP,  the succesuve terms have the
same common difference. So, B-A = C-B.

Therefore 2B =
A+C.

Since 2^A, 2^B and 2^C are in GP,  the successive tems
have the same common ratio. Therefore  2^B/2^A = 2^C / 2^A
.

(2^B)^2 = 2^A * 2^c

2^2B =
2^(A+C).

Bases being same, we equate the
powers.

2^2B = A+C.

Similarly
for x^A , x^B, and x^C ,

(x^B)^2 =
x^A*x^C

x^(2B) = x^(A+C) holds as 2B =
A+C.

Therefore for any integers in A,B and C  in A P ,  we
can choose a base x. Then  x^A, x^B and x^C  are  in
GP.

Example:

We take x = 7. 
A= 3,   B = 5 and C = 7.

7^3 , 7^5 and 7^7 are in GP with
7^2 as common ratio.