I found these formulas by playing around. I don’t know how to prove them with math.
By trial and error I played around and looked for patterns in numbers. I compared calculations with a spreadsheet.
Then, when I found the formula, my storage media crashed.
I suppose one way to show the source of one’s work is to show corrections. The sum of k^N had to be corrected to k/(k-1) rather than k/(k+1).