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.

Question

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.

mr.math · Answer

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 $g(n)=m$.

anonymous · Answer

i really want to see the solution for this problem :)