Chứng minh rằng
\(\frac{\sqrt[2016]{9}+\sqrt[2016]{16}+\sqrt[2016]{25}}{\sqrt[2016]{12}+\sqrt[2016]{15}+\sqrt[2016]{20}}>\frac{\sqrt[2017]{12}+\sqrt[2017]{15}+\sqrt[2017]{20}}{\sqrt[2017]{9}+\sqrt[2017]{16}+\sqrt[2017]{25}}\)
K
Khách
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Những câu hỏi liên quan
VT
1
NV
Nguyễn Việt Lâm
Giáo viên
23 tháng 5 2020
\(\frac{\sqrt{2016}^2}{\sqrt{2017}}+\frac{\sqrt{2017}^2}{\sqrt{2016}}\ge\frac{\left(\sqrt{2016}+\sqrt{2017}\right)^2}{\sqrt{2017}+\sqrt{2016}}=\sqrt{2016}+\sqrt{2017}\)
Dấu "=" ko xảy ra nên \(\frac{2016}{\sqrt{2017}}+\frac{2017}{\sqrt{2016}}>\sqrt{2016}+\sqrt{2017}\)
14 tháng 6 2017
a )\(\sqrt{6+\sqrt{8}+\sqrt{12}+\sqrt{24}}\)
=\(\sqrt{2+3+1+2\sqrt{2.1+2\sqrt{3}.1+2\sqrt{2}.\sqrt{3}}}\)
=\(\sqrt{\left(\sqrt{2}+\sqrt{3}+1\right)^2}\)
=\(\sqrt{2}+\sqrt{3}+1\)
15 tháng 6 2018
\(\frac{2016}{\sqrt{2016}}=\sqrt{2016}\)
\(\frac{2017}{\sqrt{2017}}=\sqrt{2017}\)
=> Bằng nhau
16 tháng 6 2018
\(\frac{2016}{\sqrt{2017}}+\frac{2017}{\sqrt{2016}}-\sqrt{2016}-\sqrt{2017}=\left(\frac{2016}{\sqrt{2017}}-\sqrt{2017}\right)+\left(\frac{2017}{\sqrt{2016}}-\sqrt{2016}\right)\)
\(=\frac{2016-2017}{\sqrt{2017}}+\frac{2017-2016}{\sqrt{2016}}=\frac{1}{\sqrt{2016}}-\frac{1}{\sqrt{2017}}\)
vì \(2016< 2017\Rightarrow\sqrt{2016}< \sqrt{2017}\Rightarrow\frac{1}{\sqrt{2016}}>\frac{1}{\sqrt{2017}}\Rightarrow\frac{1}{\sqrt{2016}}-\frac{1}{\sqrt{2017}}>0\)
\(\Rightarrow\frac{2016}{\sqrt{2017}}+\frac{2017}{\sqrt{2016}}-\sqrt{2016}-\sqrt{2017}>0\Rightarrow\frac{2016}{\sqrt{2017}}+\frac{2017}{\sqrt{2016}}>\sqrt{2016}+\sqrt{2017}\)
ML
16 tháng 5 2018
\(A=\frac{2017-2016+2017\sqrt{2016}-2016\sqrt{2017}}{\sqrt{2016}+\sqrt{2017}+\sqrt{2016.2017}}\)
= \(\frac{\left(\sqrt{2017}-\sqrt{2016}\right)\left(\sqrt{2017}+\sqrt{2016}\right)+\sqrt{2016.2017}\left(\sqrt{2017}-\sqrt{2016}\right)}{\sqrt{2016}+\sqrt{2017}+\sqrt{2016.2017}}\)
= \(\frac{\left(\sqrt{2017}-\sqrt{2016}\right)\left(\sqrt{2016}+\sqrt{2017}+\sqrt{2016.2017}\right)}{\sqrt{2016}+\sqrt{2017}+\sqrt{2016.2017}}\)
= \(\sqrt{2017}-\sqrt{2016}\)
17 tháng 8 2017
Ta có:
\(\frac{1-\sqrt{n}+\sqrt{n+1}}{1+\sqrt{n}+\sqrt{n+1}}=\frac{\left(1-\sqrt{n}+\sqrt{n+1}\right)^2}{\left(1+\sqrt{n}+\sqrt{n+1}\right)\left(1-\sqrt{n}+\sqrt{n+1}\right)}=\frac{2n+2-2\sqrt{n}+2\sqrt{n+1}-2\sqrt{n\left(n+1\right)}}{2\left(1+\sqrt{n+1}\right)}\)
\(=\frac{\left[2n+2-2\sqrt{n}+2\sqrt{n+1}-2\sqrt{n\left(n+1\right)}\right]\left(1-\sqrt{n+1}\right)}{2\left(1+\sqrt{n+1}\right)\left(1-\sqrt{n+1}\right)}=\frac{-2n\sqrt{n+1}+2n\sqrt{n}}{-2n}=\sqrt{n+1}-\sqrt{n}\)
Suy ra:
\(Q=\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{2017}-\sqrt{2016}=\sqrt{2017}-\sqrt{2}< \sqrt{2017}-1=R\)
Vậy Q < R.
WR
7 tháng 7 2017
Với mọi \(n\in N.\)ta có:
\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)^2n-n^2\left(n+1\right)}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}.\)Do đó
\(P=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2016}}-\frac{1}{\sqrt{2017}}.=1-\frac{1}{\sqrt{2017}}=\frac{\sqrt{2017}-1}{\sqrt{2017}}.\)
14 tháng 6 2017
a,\(\sqrt{6+\sqrt{8}+\sqrt{12}+\sqrt{24}}\\ =\sqrt{2+3+1+2\sqrt{2}.1+2\sqrt{3}.1+2\sqrt{2}.\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{2}+\sqrt{3}+1\right)^2}=\sqrt{2}+\sqrt{3}+1\)
24 tháng 9 2016
\(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2016}}-\frac{1}{\sqrt{2017}}=1-\frac{1}{\sqrt{2007}}=\frac{\sqrt{2007}-1}{\sqrt{2007}}\)
26 tháng 11 2021
a.
\(x=9-\dfrac{1}{\sqrt{\dfrac{9-4\sqrt{5}}{4}}}+\dfrac{1}{\sqrt{\dfrac{9+4\sqrt{5}}{4}}}\\ x=9-\dfrac{1}{\dfrac{\sqrt{5}-2}{2}}+\dfrac{1}{\dfrac{\sqrt{5}+2}{2}}\\ x=9-\left(\dfrac{2}{\sqrt{5}-2}-\dfrac{2}{\sqrt{5}+2}\right)=9-8=1\\ \Rightarrow f\left(x\right)=f\left(1\right)=\left(1-1+1\right)^{2016}=1\)
26 tháng 11 2021
c.
\(=\sin x\cdot\cos x+\dfrac{\sin^2x}{1+\dfrac{\cos x}{\sin x}}+\dfrac{\cos^2x}{1+\dfrac{\sin x}{\cos x}}\\ =\sin x\cdot\cos x+\dfrac{\sin^2x}{\dfrac{\sin x+\cos x}{\sin x}}+\dfrac{\cos^2x}{\dfrac{\sin x+\cos x}{\cos x}}\\ =\sin x\cdot\cos x+\dfrac{\sin^3x}{\sin x+\cos x}+\dfrac{\cos^3x}{\sin x+\cos x}\\ =\sin x\cdot\cos x+\dfrac{\left(\sin x+\cos x\right)\left(\sin^2x-\sin x\cdot\cos x+\cos^2x\right)}{\sin x+\cos x}\\ =\sin x\cdot\cos x-\sin x\cdot\cos x+\sin^2x+\cos^2x\\ =1\)