Let the first term be x and the common difference be d.

So, a = x + (p - 1)d

b = x + (q - 1)d

c = x + (r - 1)d

So, a - b = x + (p - 1)d - x - (q - 1)d = (p - q)d

b - c = x + (q - 1)d - x - (r - 1)d = (q - r)d

c - a = x + (r - 1)d - x - (p - 1)d = (r - p)d

Now, a(q - r) + b(r - p) + c(p - q)

= (1/d) * [a(q - r)d + b(r - p)d + c(p - q)d]

= (1/d) * [a(b - c) + b(c - a) + c(a - b)]

= (1/d) * [ab - ac + bc - ab + ac - bc]

= (1/d) * 0

= 0