# Mathematica, 29 bytes #

    #=={}||Equal@@(Most@#+Rest@#)&

A port of Luis Mendo's MATL algorithm. Unnamed function taking a list of numbers (or even more general objects) and returning `True` or `False`. Tests whether sums of consecutive elements are all equal. Unfortunately `Most` and `Rest` choke on the empty list, so that has to be tested separately.

# Mathematica, 33 bytes #

    Differences[#,1,2]~MatchQ~{0...}&

Unnamed function taking a list of numbers (or even more general objects) and returning `True` or `False`. The function `Differences[#,1,2]` takes the differences, not of *consecutive* pairs of integers, but pairs of integers at distance two apart. Then we just check whether the resulting list has nothing other than zeros in it.

As a bonus, for one more byte (change the `2` to `#2`), we get a function that inputs a list of integers and another positive integer `#2`, and checks whether the input list is the result of interleaving `#2` constant sequences periodically with one another. For example,

    Differences[#,1,#2]~MatchQ~{0...}&[{1,2,3,4,5,1,2,3,4,5,1,2},5]

evaluates to `True`.