Đáp án:\(\left[ \begin{array}{l}\sqrt{n+2\sqrt{n-1}}+\sqrt{n-2\sqrt{n-1}}=2\sqrt{n-1}(n\ge2)\\\sqrt{n+2\sqrt{n-1}}+\sqrt{n-2\sqrt{n-1}}=2(1\le n \le 2)\end{array} \right.\)
Giải thích các bước giải:
Đặt `A=\sqrt{n+2\sqrt{n-1}}+\sqrt{n-2\sqrt{n-1}}(n>=1)`
`=\sqrt{n-1+2\sqrt{n-1}+1}+\sqrt{n-1-2\sqrt{n-1}+1}`
`=\sqrt{(\sqrt{n-1}+1)^2}+\sqrt{(\sqrt{n-1}-1)^2}`
`=|\sqrt{n-1}+1|+|\sqrt{n-1}-1|`
`=\sqrt{n-1}+1+|\sqrt{n-1}-1|`
Nếu `n>=2=>n-1>=1`
`=>\sqrt{n-1}>=1`
`=>\sqrt{n-1}-1>=0`
`=>|\sqrt{n-1}-1|=\sqrt{n-1}-1`
`=>A=\sqrt{n-1}+1+\sqrt{n-1}-1=2\sqrt{n-1}`
Nếu `1<=n<=2=>n-1<=1`
`=>\sqrt{n-1}<=1`
`=>\sqrt{n-1}-1<=0`
`=>|\sqrt{n-1}-1|=1-\sqrt{n-1}`
`=>A=\sqrt{n-1}+1+1-\sqrt{n-1}=2`
