The mathematical expression
But what if you don’t understand what “+ … +” means? Sure, most people understand what that means intuitively, but how could you define “…”?
From a logical point of view, you could instead make a recursive definition
Most of the time, when I want to prove a summation, I prove it by induction anyway, so this definition is actually helpful. Suppose we want to prove
Supposing the summation to work for , for the case we have
where the first equation comes from the recursive definition of the summation, the second by hypothesis, and the third by algebra.