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moneybird
 3 years ago
Suppse that \[f(n)=2n\lfloor \frac{1+\sqrt{8n7}}{2} \rfloor\] and \[g(n)=2n\lfloor \frac{1+\sqrt{8n7}}{2} \rfloor\] 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.
moneybird
 3 years ago
Suppse that \[f(n)=2n\lfloor \frac{1+\sqrt{8n7}}{2} \rfloor\] and \[g(n)=2n\lfloor \frac{1+\sqrt{8n7}}{2} \rfloor\] 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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Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.0I 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 \(g(n)=m\).

mukushla
 3 years ago
Best ResponseYou've already chosen the best response.0i really want to see the solution for this problem :)
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