Ta có:

\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n\left(n+1\right)}\left(\sqrt{n+1}+\sqrt{n}\right)}\)

\(=\frac{\left(\sqrt{n+1}-\sqrt{n}\right)}{\sqrt{n\left(n+1\right)}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)

Thế vô bài toán được

\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{2016\sqrt{2015}+2015\sqrt{2016}}\)

\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}\)

\(=1-\frac{1}{\sqrt{2016}}\)

Xét: \(\sqrt{1+n^2+\frac{n^2}{\left(n+1\right)^2}}=\sqrt{\frac{\left(n+1\right)^2+n^2\left(n+1\right)^2+n^2}{\left(n+1\right)^2}}\) (với \(n\inℕ\))

\(=\sqrt{\frac{n^2+2n+1+n^4+2n^3+n^2+n^2}{\left(n+1\right)^2}}\)

\(=\sqrt{\frac{n^4+n^2+1+2n^3+2n^2+2n}{\left(n+1\right)^2}}\)

\(=\sqrt{\frac{\left(n^2+n+1\right)^2}{\left(n+1\right)^2}}=\frac{n^2+n+1}{n+1}=n+\frac{1}{n+1}\)

Áp dụng vào ta tính được: \(\sqrt{1+2015^2+\frac{2015^2}{2016^2}}+\frac{2015}{2016}=2015+\frac{1}{2016}+\frac{2015}{2016}\)

\(=2015+1=2016\)

Khi đó: \(\sqrt{x^2-2x+1}+\sqrt{x^2-4x+4}=2016\)

\(\Leftrightarrow\left|x-1\right|+\left|x-2\right|=2016\)

Đến đây xét tiếp các TH nhé, ez rồi:))

chẳng biết đúng ko,mới lớp 5

\(\sqrt{x^2-2x+1}+\sqrt{x^2-4x+4}=\sqrt{1+2015^2+\frac{2015^2}{2016^2}}+\frac{2015}{2016}\)

\(\sqrt{x^2}-\sqrt{2x}+\sqrt{1}+\sqrt{x^2}-\sqrt{4x}+\sqrt{4}=\sqrt{1}+\sqrt{2015^2}+\sqrt{\frac{2015^2}{2016^2}}+\frac{2015}{2016}\)

\(\sqrt{x^2}-\sqrt{6x}+3=1+2015+\frac{2015}{2016}+\frac{2015}{2016}\)

\(x-\sqrt{6x}=1+\frac{2015}{1+2016+2016}-3\)

\(x-\sqrt{6x}=2-\frac{2015}{4033}\)

\(x-\sqrt{6x}=\frac{6051}{4033}\)

pt <=>\(\sqrt{\left(x-1\right)^2}+\sqrt{\left(x-2\right)^2}=\sqrt{1+2.2015+2015^2-2.2015+\frac{2015^2}{2016^2}}+\frac{2015}{2016}\)

\(\Leftrightarrow\left(x-1\right)+\left(x-2\right)=\sqrt{2016^2-2.2016.\frac{2015}{2016}+\frac{2015^2}{2016^2}}+\frac{2015}{2016}\)

\(\Leftrightarrow2x-3=\sqrt{\left(2016-\frac{2015}{2016}\right)^2}+\frac{2015}{2016}\)

\(\Leftrightarrow2x-3=2016-\frac{2015}{2016}+\frac{2015}{2016}\)

\(\Leftrightarrow2x-3=2016\)

\(\Leftrightarrow2x=2019\)

\(\Leftrightarrow x=\frac{2019}{2}\)

\(\frac{1}{\left(k+1\right)\sqrt{k}+k\left(\sqrt{k+1}\right)}=\frac{\left(k+1\right)\sqrt{k}-k\left(\sqrt{k+1}\right)}{\left(k+1\right)^2k-k^2\left(k+1\right)}\) 

                                                 =\(\frac{\left(k+1\right)\sqrt{k}-k\left(\sqrt{k+1}\right)}{\left(k+1\right)k\left(k+1-k\right)}\)

                                                    =\(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\)

áp dụng vào biểu thức ta có\(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}\)

                       =\(1-\frac{1}{\sqrt{2016}}\)

   đến đây cậu tự giải nốt nhé

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mình ko có sách đấy 

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giúp đi

Giải tổng quát nha : 

\(\frac{1}{x\sqrt{x+1}+\left(x+1\right)\sqrt{x}}=\frac{1}{\sqrt{x\left(x+1\right)}\left(\sqrt{x}+\sqrt{x+1}\right)}=\frac{\sqrt{x+1}-\sqrt{x}}{\sqrt{x\left(x+1\right)}}=\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\)

\(\sqrt{1+n^2+\frac{n^2}{\left(n+1\right)^2}}=\sqrt{\left(1+n-\frac{n}{n+1}\right)^2}=1+n-\frac{n}{n+1}\text{ }\left(n>0\right)\)

\(P==1+2015-\frac{2015}{2016}+\frac{2015}{2016}=2016\)

\(\left(1+n-\frac{n}{n+1}\right)^2=1+n^2+\frac{n^2}{\left(n+1\right)^2}+2\left(n-\frac{n}{n+1}-\frac{n^2}{n+1}\right)\)

\(=1+n^2+\frac{n^2}{\left(n+1\right)^2}+2.\frac{n^2+n-n-n^2}{n+1}\)

\(=1+n^2+\frac{n^2}{\left(n+1\right)^2}\)

Sao mr. Lazy làm nhiều thế?