`@` `\color{Silver}{\text{πk2}`

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`a)` Ta có:

`1/(\sqrt{1}+\sqrt{3})>1/(\sqrt{3}+\sqrt{5})`

`1/(\sqrt{5}+\sqrt{7})>1/(\sqrt{7}+\sqrt{9})`

`...`

`1/(\sqrt{97}+\sqrt{99})>1/(\sqrt{99}+\sqrt{101})`

`⇒2·(1/(\sqrt{1}+\sqrt{3})+1/(\sqrt{5}+\sqrt{7})+1/(\sqrt{9}+\sqrt{11})+...+1/(\sqrt{97}+\sqrt{99}))`

`>1/(\sqrt{1}+\sqrt{3})+1/(\sqrt{3}+\sqrt{5})+1/(\sqrt{5}+\sqrt{7})+1/(\sqrt{7}+\sqrt{9})+1/(\sqrt{9}+\sqrt{11})+...+1/(\sqrt{97}+\sqrt{99})+1/(\sqrt{99}+\sqrt{101})`

`=1/2·(2/(\sqrt{1}+\sqrt{3})+2/(\sqrt{3}+\sqrt{5})+2/(\sqrt{5}+\sqrt{7})+2/(\sqrt{7}+\sqrt{9})+2/(\sqrt{9}+\sqrt{11})+...+2/(\sqrt{97}+\sqrt{99})+2/(\sqrt{99}+\sqrt{101}))`
`=1/2·((3-1)/(\sqrt{3}+\sqrt{1})+(5-3)/(\sqrt{5}+\sqrt{3})+(7-5)/(\sqrt{7}+\sqrt{5})+(9-7)/(\sqrt{9}+\sqrt{7})+(11-9)/(\sqrt{11}+\sqrt{9})+...+(99-97)/(\sqrt{99}+\sqrt{97})+(101-99)/(\sqrt{101}+\sqrt{99}))`

`=1/2·[[(\sqrt{3}-\sqrt{1})(\sqrt{3}+\sqrt{1})]/(\sqrt{3}+\sqrt{1})+[(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})]/(\sqrt{5}+\sqrt{3})+[(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{5})]/(\sqrt{7}+\sqrt{5})+[(\sqrt{9}-\sqrt{7})(\sqrt{9}+\sqrt{7})]/(\sqrt{9}+\sqrt{7})+[(\sqrt{11}-\sqrt{9})(\sqrt{11}+\sqrt{9})]/(\sqrt{11}+\sqrt{9})+...+[(\sqrt{99}-\sqrt{97})(\sqrt{99}+\sqrt{97})]/(\sqrt{99}+\sqrt{97})+[(\sqrt{101}-\sqrt{99})(\sqrt{101}+\sqrt{99})]/(\sqrt{101}+\sqrt{99})]`

`=1/2·(\sqrt{3}-\sqrt{1}+\sqrt{5}-\sqrt{3}+\sqrt{7}-\sqrt{5}+\sqrt{9}-\sqrt{7}+\sqrt{11}-\sqrt{9}+...+\sqrt{99}-\sqrt{97}+\sqrt{101}-\sqrt{99})`

`=1/2·(\sqrt{101}-1)`

`>1/2·(\sqrt{100}-1)`

`=1/2·(10-1)`

`=1/2·9`

`=9/2`

`⇒2·(1/(\sqrt{1}+\sqrt{3})+1/(\sqrt{5}+\sqrt{7})+1/(\sqrt{9}+\sqrt{11})+...+1/(\sqrt{97}+\sqrt{99}))>9/2`

`⇒1/(\sqrt{1}+\sqrt{3})+1/(\sqrt{5}+\sqrt{7})+1/(\sqrt{9}+\sqrt{11})+...+1/(\sqrt{97}+\sqrt{99})>9/4`

-------------------------

Ta có:

`1/(\sqrt{1}+\sqrt{2})>1/(\sqrt{2}+\sqrt{3})`

`1/(\sqrt{3}+\sqrt{4})>1/(\sqrt{4}+\sqrt{5})`

`...`

`1/(\sqrt{79}+\sqrt{80})>1/(\sqrt{80}+\sqrt{81})`

`⇒2·(1/(\sqrt{1}+\sqrt{2})+1/(\sqrt{3}+\sqrt{4})+1/(\sqrt{5}+\sqrt{6})+...+1/(\sqrt{79}+\sqrt{80}))`

`>1/(\sqrt{1}+\sqrt{2})+1/(\sqrt{2}+\sqrt{3})+1/(\sqrt{3}+\sqrt{4})+1/(\sqrt{4}+\sqrt{5})+1/(\sqrt{5}+\sqrt{6})+...+1/(\sqrt{79}+\sqrt{80})+1/(\sqrt{80}+\sqrt{81})`

`=(2-1)/(\sqrt{2}+\sqrt{1})+(3-2)/(\sqrt{3}+\sqrt{2})+(4-3)/(\sqrt{4}+\sqrt{3})+(5-4)/(\sqrt{5}+\sqrt{4})+(6-5)/(\sqrt{6}+\sqrt{5})+...+(80-79)/(\sqrt{80}+\sqrt{79})+(81-80)/(\sqrt{81}+\sqrt{80})`

`=[(\sqrt{2}-\sqrt{1})(\sqrt{2}+\sqrt{1})]/(\sqrt{2}+\sqrt{1})+[(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})]/(\sqrt{3}+\sqrt{2})+[(\sqrt{4}-\sqrt{3})(\sqrt{4}+\sqrt{3})]/(\sqrt{4}+\sqrt{3})+[(\sqrt{5}-\sqrt{4})(\sqrt{5}+\sqrt{4})]/(\sqrt{5}+\sqrt{4})+[(\sqrt{6}-\sqrt{5})(\sqrt{6}+\sqrt{5})]/(\sqrt{6}+\sqrt{5})+...+[(\sqrt{80}-\sqrt{79})(\sqrt{80}+\sqrt{79})]/(\sqrt{80}+\sqrt{79})+[(\sqrt{81}-\sqrt{80})(\sqrt{81}+\sqrt{80})]/(\sqrt{81}+\sqrt{80})`

`=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+\sqrt{5}-\sqrt{4}+\sqrt{6}-\sqrt{5}+...+\sqrt{80}-\sqrt{79}+\sqrt{81}-\sqrt{80}`

`=\sqrt{81}-\sqrt{1}`

`=9-1`

`=8`

`⇒2·(1/(\sqrt{1}+\sqrt{2})+1/(\sqrt{3}+\sqrt{4})+1/(\sqrt{5}+\sqrt{6})+...+1/(\sqrt{79}+\sqrt{80}))>8`

`⇒1/(\sqrt{1}+\sqrt{2})+1/(\sqrt{3}+\sqrt{4})+1/(\sqrt{5}+\sqrt{6})+...+1/(\sqrt{79}+\sqrt{80})>4`