I think it was Gauss who taught us that 1+2+...+n = n(n+1)/2 for natural numbers
We can easily verify this with a proof by induction.
However, what if I would like to find the sum of all the natural numbers between any two given natural numbers (not just between 1 and a given natural number)? And from there, what if I wanted to generalize to all integers?
Well, using Gauss' reasoning it's not hard to develop an equation that would satisfy this.
For example, I could say that
For all x and all y,
The sum between x and y (inclusive: implying that if x and y are equal it will return that value) is:
(x+y)(abs(abs(x)-abs(y))+1)/2 where x,y are elements of Z.
I may have typed that in wrong; I'm not sure. But it's not really relevant. What I'm really wondering is how I would go about proving something like this.
My first instinct is that I could just do induction on the process of induction. That is, say we were just sticking with a proof within the natural numbers, if I prove with induction that it is equal to the sum of the values from 1 to y, that is a proof of the base case where x = 1. Then I could just do the inductive step on that...proving 2 to y, 3 to y, and so on. And I could do this in such a way that it wouldn't matter whether or not x>y (ie: doing the induction in both directions: doing the induction on the induction with the base case of 1 to x, and doing the induction on the induction with the base case of 1 to y). But there are other ways (more simple ways I think even) that I could go about making sure that it wouldn't matter whether x>y.
Nonetheless, my professor says that this method of applying induction on induction will not work. She also says I would have to use function spaces for this proof (which I haven't learned yet). Is my professor correct? Why or why not? Also, are there any other methods for the proof that anyone knows?
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