Chứng minh: \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}=|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}|\)
Áp dụng tính: M= \(\sqrt{1+999^2+\frac{999^2}{1000^2}}+\frac{999}{1000}\)
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2 tháng 9 2018
\(VT=\sqrt{\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right)^2-\left(\frac{2}{ab}-\frac{2}{a\left(a+b\right)}-\frac{2}{b\left(a+b\right)}\right)}\)
\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right)^2-\frac{2\left(a+b\right)-2b-2a}{ab\left(a+b\right)}}\)
\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right)^2}=\left|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right|=VP\)
Áp dụng tính M: \(M=\sqrt{1+999^2+\frac{999^2}{1000^2}}+\frac{999}{1000}\)
\(M=999.\sqrt{\frac{1}{999^2}+\frac{1}{1^2}+\frac{1}{\left(999+1\right)^2}}+\frac{999}{1000}\)
\(M=999.\left(\frac{1}{1}+\frac{1}{999}-\frac{1}{1000}\right)+\frac{999}{1000}\)
\(M=999+1-\frac{999}{1000}+\frac{999}{1000}=1000\)
Vậy M=1000.
28 tháng 8 2020
đầu bài phải là: cmr: \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}=\left|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right|\)chì bn???
Giải:
\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}-2.\left(\frac{b+a-a-b}{ab.\left(a+b\right)}\right)}\)
\(=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}-2.\left(\frac{1}{a.\left(a+b\right)}+\frac{1}{b.\left(a+b\right)}-\frac{1}{ab}\right)}\)
\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right)^2}=\left|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right|\)
=> đpcm
AD: \(\sqrt{1+999^2+\frac{999^2}{1000^2}}+\frac{999}{1000}=\left|1+999-\frac{999}{1000}\right|+\frac{999}{1000}\)
\(=1000-\frac{999}{1000}+\frac{999}{1000}=1000\)
6 tháng 8 2020
- Gỉa sử \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}=\left|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right|\)
=> \(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}=\left(\left|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right|\right)^2\)
=> \(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}+\frac{2}{ab}-\frac{2}{b\left(a+b\right)}-\frac{2}{a\left(a+b\right)}\)
=> \(\frac{2}{ab}-\frac{2}{b\left(a+b\right)}-\frac{2}{a\left(a+b\right)}=0\)
=> \(\frac{a+b}{ab\left(a+b\right)}-\frac{a}{ab\left(a+b\right)}-\frac{b}{ab\left(a+b\right)}=0\)
=> \(\frac{a+b-a-b}{ab\left(a+b\right)}=\frac{0}{ab\left(a+b\right)}=0\) (Luôn đúng )
Vậy ....
- Áp dụng : \(M=\sqrt{1+999^2+\frac{999^2}{1000^2}}+\frac{999}{1000}\)
=> \(M=\sqrt{1+999^2+\frac{999^2}{\left(1+999\right)^2}}+\frac{999}{1000}\) ( với \(a=1,b=999\) )
=> \(M=1+999-\frac{999}{1000}+\frac{999}{1000}=1000\)
TT
0
7 tháng 4 2015
\(A=\frac{1}{\sqrt{2.1}\left(\sqrt{2}+\sqrt{1}\right)}+\frac{1}{\sqrt{2.3}\left(\sqrt{3}+\sqrt{2}\right)}+\frac{1}{\sqrt{3.4}\left(\sqrt{4}+\sqrt{3}\right)}+...+\frac{1}{\sqrt{999.1000}\left(\sqrt{1000}+\sqrt{999}\right)}\)
\(A=\frac{\sqrt{2}-\sqrt{1}}{\sqrt{2.1}\left(2-1\right)}+\frac{\sqrt{3}-\sqrt{2}}{\sqrt{2.3}\left(3-2\right)}+\frac{\sqrt{4}-\sqrt{3}}{\sqrt{3.4}\left(4-3\right)}+...+\frac{\sqrt{1000}-\sqrt{999}}{\sqrt{999.1000}\left(1000-999\right)}\)
\(A=\frac{\sqrt{2}}{\sqrt{2.1}}-\frac{\sqrt{1}}{\sqrt{2.1}}+\frac{\sqrt{3}}{\sqrt{2.3}}-\frac{\sqrt{2}}{\sqrt{2.3}}+\frac{\sqrt{4}}{\sqrt{3.4}}-\frac{\sqrt{3}}{\sqrt{3.4}}+...+\frac{\sqrt{1000}}{\sqrt{999.1000}}-\frac{\sqrt{999}}{\sqrt{1000.999}}\)
\(A=\frac{1}{1}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{999}}-\frac{1}{\sqrt{1000}}\)
\(A=\frac{1}{1}-\frac{1}{\sqrt{1000}}=\frac{\sqrt{1000}-1}{\sqrt{1000}}=\frac{10\sqrt{10}-1}{10\sqrt{10}}\)