Hello there, for a while I found a weird way to calculate the difference of squares. I included 2 pics with this post, the one with all the math work is the weird way I found. Today I refined it to the formula that is in the other picture. I haven't found anything online that points this out as a phenomenon, I'm not claiming ownership. But if this is the first time this has been uncovered then awesome. I am mainly trying to find a proof for this, thank you.
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In the first picture, it looks like you are claiming n^2 - (n-3)^2 = n + 2((n-1)+(n-2)) + (n-3)
And sure enough, thats true if you expand it out!
in the second picture, im not quite sure whats going on...
a^2 - b^2 = 1/(a-b) * [ ∫ (a^2 - b^2)dx from b to a]
a^2 - b^2 = 1/(a-b) * [(a^2 - b^2)*x from b to a]
a^2 - b^2 = 1/(a-b) * [(a^2 - b^2)(a-b)] a^2 - b^2 = (a-b)/(a-b) * (a^2 - b^2)
a^2 - b^2 = a^2 - b^2
So the equation holds, but I'm not sure how it's a proof of the first picture. basically youre saying the average value of the function f(x) = a^2 - b^2 from b to a is equal to a^2 - b^2, but a^2 - b^2 is a constant function, so it has the same average value everywhere.
Maybe youre trying to claim something about n^2 - (n-k)^2 being related to binomial coefficients, not sure.