Consider any set of three adjacent elements in the same period on the periodic table. for which characteristic is the average for the three elements always equal to the value of the middle element?

The answer is the atomic number.

The elements in the periodic table are arranged in increasing order of atomic number.

Thre adjacent elements in the same period (row) of the periodic table will have consecutive atomic numbers.

If the atomic number of the first of those three elements is n, next is n +1 and next is n +2.

The average of those three numbers is [n + (n+1) + (n + 2)] / 3 = [3n + 3] / 3 = n + 1. As you see n + 1 is the atomic number of the middle element, which proves the answer.

Compound with sulfer  is bright red