## moneybird Group Title 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. 2 years ago 2 years ago

I feel lazy to try it out now, but I have an idea for you that you might want to try. Divide your proof into two parts. i) The first part shows that $$g(n)\ne f(n)$$ $$\forall n\in \mathbb{N}$$. ii) The second part shows that if $$\forall n\in \mathbb{N}$$, $$f(n)\ne m$$ for some integer m, then $$\exists n \in \mathbb{N}$$ such that $$g(n)=m$$.