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\(A=\frac{2010}{1}+\frac{2009}{2}+...+\frac{2}{2009}+\frac{1}{2010}\)
\(A=1+\left(\frac{2009}{2}+1\right)+...+\left(\frac{2}{2009}+1\right)+\left(\frac{1}{2010}+1\right)\)
\(A=\frac{2011}{2011}+\frac{2011}{2}+...+\frac{2011}{2009}+\frac{2011}{2010}\)
\(A=\frac{2011}{2}+...+\frac{2011}{9}+\frac{2011}{10}+\frac{2011}{11}\)
\(A=2011.\left(\frac{1}{2}+...+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}\right)\)
\(A=2011.B\)
Nên : \(\frac{A}{B}=\frac{2011.B}{B}=2011\)
Vậy \(\frac{A}{B}=2011\)
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\(\frac{2010}{1}+\frac{2009}{2}+...+\frac{1}{2010}\)
\(=\left(1+1+...+1\right)+\frac{2009}{2}+...+\frac{1}{2010}\)
\(=\left(\frac{2009}{2}+1\right)+\left(\frac{2008}{3}+1\right)+...+\left(\frac{1}{2010}+1\right)+1\)
\(=\frac{2011}{2}+\frac{2011}{3}+...+\frac{2011}{2010}+\frac{2011}{2011}\)
\(=2011.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2010}+\frac{1}{2011}\right)\)
Ta thấy tử số gấp 2011 lần mẫu số nên phép tính này=2011
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Ta có: \(C=\frac{\frac{2010}{1}+\frac{2009}{2}+\frac{2008}{3}+...+\frac{1}{2010}}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2011}}\)
Đặt \(A=\frac{2010}{1}+\frac{2009}{2}+\frac{2008}{3}+...+\frac{1}{2010}\)
\(A=\frac{2010}{1}+1+\frac{2009}{1}+1+\frac{2008}{1}+1+...+\frac{1}{2010}+1-2010\)
\(=\frac{2011}{1}+\frac{2011}{2}+\frac{2011}{3}+...+\frac{2011}{2010}-\frac{2011.2010}{2011}\)
\(=2011\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2010}-\frac{2010}{2011}\right)\)
Đặt \(B=\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2011}\)
\(B=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2011}-1\)
\(=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2010}-\frac{2010}{2011}\)
Ta có: \(C=\frac{A}{B}=2011\)(lấy A-B)
Ta có :
\(2010A=\dfrac{2010^{2012}+2010}{2010^{2012}+1}=\dfrac{2010^{2012}+1+2009}{2010^{2012}+1}=1+\dfrac{2009}{2010^{2012}+1}\)
\(2010B=\dfrac{2010^{2011}+2010}{2010^{2011}+1}=\dfrac{2010^{2011}+1+2009}{2010^{2011}+1}=1+\dfrac{2009}{2010^{2011}+1}\)
Vì \(1+\dfrac{2009}{2010^{2012}+1}< 1+\dfrac{2009}{2010^{2011}+1}\Rightarrow A< B\)
~ Học tốt ~
a/\(\frac{\left(2^3.5.7\right).\left(5^2.7^3\right)}{\left(2.5.7^2\right)^2}\)
=\(\frac{2^3.5^3.7^4}{2^2.5^2.7^4}\)
=2.5
=10
a) A= 1/2010+1+2/2009+1+3/2008+1+...+2009/2+1+1
= 2011/2010+20011/2009+2011/2008+...+2011/2+2011/2011
= 2011(1/2+1/3+1/4+...+1/2011)
Ta có: B= 1/2+1/3+1/4+...+1/2011
suy ra A/B= 2011
=1/2010