Why is the inverse of a gradient -1/x?

Because the value, x and its inverse, -1/x should multiply together to give -1. This is proof that two lines are perpendicular (or having the same magnitude of gradient but opposite signs +/- ).

SR

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Show the sum from n=0 to 200 of x^n given that x is not 1, is (1-x^201)/(1-x) hence find the sum of 1+2(1/2)+3(1/2)^2+...+200(1/2)^199


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