What's the name of the theorem? Is it Levi's?

Question

Given a series of numbers forming odd element, the sum is the product of the last and the middle term.

1+2+3+4+5+6+7 = 7*4
              = 28

I have noted it as Levi's theorem, but I forgot the original reference. And I am not able to find online. Please suggest the name of the theorem and a citation. Thanks.

Answer 1

This is not any particularly special theorem. It simply the formula for the nth Triangular number, $T_n$

$$T_{n} = \sum_{k=1}^{n} k = n\cdot\frac{n+1}{2} = \frac{n}{2} \cdot (n+1)$$

Although Gauss is rumored to have "discovered" this formula, it was known back in the 5th century. That is, the 5th century BC.

Answer 2

This is the Gauss sum,

$$ 1+2+3+\ldots+n=\sum_{i=1}^n i = \frac{n(n+1)}{2}$$

for any natural number $n$, and I dare say that it is special, because it is the easiest "general" sum one could come up with, besides summing the same number over and over, which leads to multiplication. It is rumored that Gauss discovered this formula for himself in elementary school: When a math teacher was lazy, he could just let his pupils sum up the first 100 natural numbers and they would be busy for a while. Little Gauss (actually how this sum is called in Germany, too), figured out a quick way to calculate this. Nowadays this is taken as the first proof of his ingenuity. Gauss did in fact become one of the greatest mathematicians there are.

I see the English Wikipedia doesn't contain a reference to this kind of Gauss sum and MathWorld not as well, so I give you the German Wikipedia on Little Gauss and Great Gauss, too. (The latter has in fact an English variant, but it shorter.) Maybe you can translate the pages, I don't know.

Gauss himself was German (as I am) and even featured on the German currency until the Euro came along, so it is no wonder that the German literature on him is bigger.