Xét tổng quát :
\(\frac{1}{\sqrt{n}-\sqrt{n+1}}=\frac{\sqrt{n+1}+\sqrt{n}}{\left(\sqrt{n}-\sqrt{n+1}\right)\left(\sqrt{n}-\sqrt{n+1}\right)}=\frac{\sqrt{n}+\sqrt{n+1}}{n-n-1}=-\left(\sqrt{n}-\sqrt{n+1}\right)=-\sqrt{n}+\sqrt{n+1}\)
Với n = 2 ; 3 ; 4 ; .... 2015 ta có :
\(Q=-\sqrt{2}+\sqrt{3}-\sqrt{3}+\sqrt{4}+....-\sqrt{2015}+\sqrt{2016}\)
Q = \(\sqrt{2016}-\sqrt{2}\)
RGBT:
E=\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+...+\frac{1}{2015\sqrt{2014}+2014\sqrt{2015}}+\frac{1}{2016\sqrt{2015}+2015\sqrt{2016}}\)
1. Rút Gọn:
\(P=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(Q=\frac{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\)
2. Tính Tổng:
\(S=\frac{\sqrt{1}+\sqrt{2}}{1+2}+\frac{\sqrt{2}+\sqrt{3}}{2+3}+...+\frac{\sqrt{2015}+\sqrt{2016}}{2015+2016}\)
Rút gọn D, biết D=\(\frac{1}{\sqrt{2}+2}\)+ \(\frac{1}{3\sqrt{2}+2\sqrt{3}}\)+ \(\frac{1}{4\sqrt{3}+3\sqrt{4}}\)+........................+ \(\frac{1}{2016\sqrt{2015}+2015\sqrt{2016}}\)
Tính \(A=\sqrt{1}-\sqrt{2}+\frac{1}{\sqrt{3}}+\sqrt{4}-\sqrt{5}+\frac{1}{\sqrt{6}}+....+\sqrt{2014}-\sqrt{2015}+\frac{1}{\sqrt{2016}}\)
Rút gọn :
\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+....+\frac{1}{2016\sqrt{2015}+2015\sqrt{2016}.}\)
\(P=\frac{\sqrt{x}}{\sqrt{x}-1}\)
Tìm x nguyên dương thỏa:
\(P< \frac{1}{1\sqrt{2}+2\sqrt{1}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+...+\frac{1}{2015\sqrt{2016}+2016\sqrt{2015}}\)
1)Chứng minh
\(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{2015}+\sqrt{2016}}=\sqrt{2016}-1\)
2:Giải Phương trình:
\(\frac{3}{2}\sqrt{4x-8}-9\sqrt{\frac{x-2}{81}}=6\)
Tính:
\(\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{2015^2}+\frac{1}{2016^2}}\)
