Dễ dàng chứng minh điều sau bằng biến đổi tương đương:\(\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}}=\sqrt{\left(1+\frac{1}{n}-\frac{1}{n+1}\right)^2}=1+\frac{1}{n}-\frac{1}{n+1}\)
Áp dụng:
\(M=1+\frac{1}{1}-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+...+1+\frac{1}{2015}-\frac{1}{2016}\)
\(=2016-\frac{1}{2016}\)
RGBT:
E=\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+...+\frac{1}{2015\sqrt{2014}+2014\sqrt{2015}}+\frac{1}{2016\sqrt{2015}+2015\sqrt{2016}}\)
Tính:
\(\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{2015^2}+\frac{1}{2016^2}}\)
Rút gọn :
\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+....+\frac{1}{2016\sqrt{2015}+2015\sqrt{2016}.}\)
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:
\(\frac{3}{2}\sqrt{4x-8}-9\sqrt{\frac{x-2}{81}}=6\)
Rút gọn D, biết D=\(\frac{1}{\sqrt{2}+2}\)+ \(\frac{1}{3\sqrt{2}+2\sqrt{3}}\)+ \(\frac{1}{4\sqrt{3}+3\sqrt{4}}\)+........................+ \(\frac{1}{2016\sqrt{2015}+2015\sqrt{2016}}\)
CMR: \(A=\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+...+\frac{1}{2016\sqrt{2015}}<2\).
\(P=\frac{\sqrt{x}}{\sqrt{x}-1}\)
Tìm x nguyên dương thỏa:
\(P< \frac{1}{1\sqrt{2}+2\sqrt{1}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+...+\frac{1}{2015\sqrt{2016}+2016\sqrt{2015}}\)
giúp vs
1)a) n thuộc N*: rút gọn:
K = \(\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}}\)
b) tính
I = \(\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{2015^2}+\frac{1}{2016^2}}+\sqrt{1+\frac{1}{2016^2}+\frac{1}{2017^2}}\)2) A= \(\sqrt{x^2-6x+9}-\sqrt{x^2+6x+9}\)
a) rút gọn A
b) tìm x đề A=1
3) rút gọn B = \(\sqrt{x+\sqrt{2x-1}}-\sqrt{x-\sqrt{2x-1}}\)
4) tính: \(\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}\)
C= \(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
