Stima

[alunik]

Dimostrare che $ \displaystyle{\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\frac{1}{\sqrt{9}+\sqrt{11}}+...+\frac{1}{\sqrt{9997}+\sqrt{9999}}>24} $

[Chuck Schuldiner]

Razionalizzando ottengo che $ (\sqrt3+\sqrt7+...+\sqrt{9999})-(\sqrt1+\sqrt5+...+\sqrt{9997})>48 $.
Ma per QM-AM $ \sqrt{2i+1}> \frac{\sqrt{2i-1}+\sqrt{2i+3}}{2} $, da cui $ LHS> (\frac{\sqrt1}{2}+\frac{\sqrt5}{2}+\frac{\sqrt5}{2}+\frac{\sqrt9}{2}+...+\frac{\sqrt{9997}}{2}+\frac{\sqrt{10001}}{2})-(\sqrt1+\sqrt5+...+\sqrt{9997})=\frac{\sqrt{10001}-1}{2}>48 $$ \square $