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That seems complicated.
Average is (100+1)÷2= 50.5
50.5 average x 100 numbers = 5050.
You did essentially the exact same thing in a different order.
(100+1)÷2 × 100
(100+1)x100 ÷ 2
Clever though
The neat thing about math is that you can generally find several different paths to the same solution, and go with whichever is most intuitive to you.
Somewhat long story time.
Because I moved from a different country to the US they got confused what grade to put me in. And then they really goofed my class assignments. So I ended up literally only taking algebra 1 and geometry in high school. Unlike most that also got algebra 2 and possibly trig.
This set me up supery poorly for college. But I'm actually really good at math and I love math. So when I got to college, I failed miserably and dropped out after teachers were telling me this was simple stuff I should know from high school trig. How the college even let me get to that point was also insane...
Anyway years later I went back, they tested me and I needed remedial math classes. Finally this new college was doing it right. I immediately started failing my tests. The teacher actually cared to find out why. I was getting every answer correct, but I was only finishing about 50-60% of the questions. When they looked at my work they noticed I just want using the right formulas in the right places or the common shortcuts and such. At they focused on teaching me that stuff all of a sudden I was actually finishing my tests ahead of class and passing.
Math is weird.
Though sadly ime that is what teachers usually completely fail to convey. And then we wonder why so many people hate math.
I like to imagine These ppl have dark robes and candles cause I dk how they had the thoughts to do this. Just built different.
It's simply just breaking down a problem into clear, definable pieces, then finding the right tool to solve each piece in sequence. The hard part is learning all the tools you need and matching them with the right problems, but that just comes with time and practice.
I notice a lot of people get intimidated and freeze up. I've done quite a bit of tutoring, and for three most part one I can get someone to calm down and focus on little steps, suddenly everything makes more sense.
I think you'd need to prove that the average is (100+1)/2 because that's not an axiom.