Mathematics - Sequence & Series Question with Solution | TestHub

The four numbers are -1,0,1,2. Their sum is 2. Proof of uniqueness: write the terms as a-d,a,a+d,a+2d, with integer d>=1. Choosing each term gives a quadratic in a. Its discriminant is respectively 1-24d(d+1), 1-56d²-8d, 1-56d²+8d, or 1-24d(d-1). Only the fourth can be nonnegative, and only for d=1. It then gives a=0 or -1/3; integrality forces a=0. Thus the unique set is -1,0,1,2.