Exercitii usoare
Posted: Mon Dec 08, 2008 8:59 am
a)Fie numarul :
\( n=\frac{(\frac{1}{\sqrt{(2-\sqrt{2})^2}\cdot\sqrt{(3-\sqrt{2})^2}}+\frac{1}{\sqrt{(3-\sqrt{2})^2}\cdot\sqrt{(4-\sqrt{2})^2}}+...+\frac{1}{\sqrt{(99-\sqrt{2})^2}\cdot\sqrt{(100-\sqrt{2})^2}})}{\frac{98}{(2-\sqrt{2})(100-\sqrt{2})}} \)
Aratati ca \( n\in\mathb{Z} \) .
b)Fie numarul \( E=n[(1+\frac{1}{2})(1+\frac{1}{3})...(1+\frac{1}{n})-(1-\frac{1}{2})(1-\frac{1}{3})...(1-\frac{1}{n})] \) . Aratati ca \( E \) este natural pentru orice \( n\in\mathb{N} \) .
c)Fie \( S=\{n\in\mathb{N}|(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{n}+\sqrt{n+1}})\in\mathb{N}\} \) . Ce conditie trebuie sa indeplineasca n pentru ca \( S\neq\Phi \) .
\( n=\frac{(\frac{1}{\sqrt{(2-\sqrt{2})^2}\cdot\sqrt{(3-\sqrt{2})^2}}+\frac{1}{\sqrt{(3-\sqrt{2})^2}\cdot\sqrt{(4-\sqrt{2})^2}}+...+\frac{1}{\sqrt{(99-\sqrt{2})^2}\cdot\sqrt{(100-\sqrt{2})^2}})}{\frac{98}{(2-\sqrt{2})(100-\sqrt{2})}} \)
Aratati ca \( n\in\mathb{Z} \) .
b)Fie numarul \( E=n[(1+\frac{1}{2})(1+\frac{1}{3})...(1+\frac{1}{n})-(1-\frac{1}{2})(1-\frac{1}{3})...(1-\frac{1}{n})] \) . Aratati ca \( E \) este natural pentru orice \( n\in\mathb{N} \) .
c)Fie \( S=\{n\in\mathb{N}|(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{n}+\sqrt{n+1}})\in\mathb{N}\} \) . Ce conditie trebuie sa indeplineasca n pentru ca \( S\neq\Phi \) .