• anonymous
Suppse that $f(n)=2n-\lfloor \frac{1+\sqrt{8n-7}}{2} \rfloor$ and $g(n)=2n\lfloor \frac{1+\sqrt{8n-7}}{2} \rfloor$ for each positive integer n. Suppose that A = {f(1); f(2); f(3); ...} and B = {g(1); g(2); g(3);...}; that is, A is the range of f and B is the range of g. Prove that every positive integer m is an element of exactly one of A or B.
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