Chứng minh `(2-sqrt(2+sqrt(2+sqrt(2+ sqrt(2)))))/ (2-sqrt(2+sqrt(2+sqrt(2)))` `<1/3` - Yêu cầu giải chi tiết giúp mình ạ.

Ta sử dụng phép biến đổi tương đương.

`(2-sqrt{2+sqrt{2+sqrt{2+sqrt2}}})/(2-sqrt{2+sqrt{2+sqrt2}})<1/3`

`<=>(6-3sqrt{2+sqrt{2+sqrt{2+sqrt2}}}-2+sqrt{2+sqrt{2+sqrt2}})/(3(2-sqrt{2+sqrt{2+sqrt2}}))<0`

`<=>(4-3sqrt{2+sqrt{2+sqrt{2+sqrt2}}}+sqrt{2+sqrt{2+sqrt2}})/(3(2-sqrt{2+sqrt{2+sqrt2}}))<0`

Vì `sqrt2<sqrt4`

`<=>sqrt{2+sqrt2}<sqrt{2+sqrt4}=2`

`<=>sqrt{2+sqrt{2+sqrt2}}<sqrt{2+2}=2`

`<=>2-sqrt{2+sqrt{2+sqrt2}}>2-2=0`

Vậy nhất thiết ta phải chứng minh `4-3sqrt{2+sqrt{2+sqrt{2+sqrt2}}}+sqrt{2+sqrt{2+sqrt2}}<0`.

Ta có:`sqrt2>sqrt{1,9881}`

`<=>sqrt2>1,41`

`<=>2+sqrt2>3,41`

`<=>sqrt{2+sqrt2}>sqrt{3,41}`

Mà `sqrt{3,41}>sqrt{3,3856}=1,84`

`<=>sqrt{2+sqrt2}>1,84`

`<=>2+sqrt{2+sqrt2}>3,84`

`<=>sqrt{2+sqrt{2+sqrt2}}>sqrt{3,84}`

Mà `sqrt{3,84}>sqrt{3,8025}=1,95`

`<=>2+sqrt{2+sqrt{2+sqrt2}}>3,95`

`<=>sqrt{2+sqrt{2+sqrt{2+sqrt2}}}>sqrt{3,95}`

Mà `sqrt{3,95}>sqrt{3,94975876}=1,9874`

`<=>sqrt{2+sqrt{2+sqrt{2+sqrt2}}}>1,9874`

`<=>3sqrt{2+sqrt{2+sqrt{2+sqrt2}}}>5,9622`

`sqrt2<sqrt{2,0164}=1,42`

`<=>2+sqrt2<3,42`

`<=>sqrt{2+sqrt2}<sqrt{3,42}`

Mà `sqrt{3,42}<sqrt{3,4225}=1,85`

`<=>2+sqrt{2+sqrt2}<3,85`

`<=>sqrt{2+sqrt{2+sqrt2}}<sqrt{3,85}`

Mà `sqrt{3,85}<sqrt{3,850032623}=1,96215`

`<=>sqrt{2+sqrt{2+sqrt2}}<1,96215<1,9622`

`<=>4+sqrt{2+sqrt{2+sqrt2}}<5,9622`

Mà `3sqrt{2+sqrt{2+sqrt{2+sqrt2}}}>5,9622`

`=>4+sqrt{2+sqrt{2+sqrt2}}-3sqrt{2+sqrt{2+sqrt{2+sqrt2}}}<0`

`=>` ta có điều phải chứng minh là `(2-sqrt{2+sqrt{2+sqrt{2+sqrt2}}})/(2-sqrt{2+sqrt{2+sqrt2}})<1/3`.

Trả lời