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Ta có: \(5A=\frac{5^{2011}+5}{5^{2011}+1}=\frac{5^{2011}+1+4}{5^{2011}+1}=1+\frac{4}{5^{2011}+16}\)
\(5B=\frac{5^{2010}+5}{5^{2010}+1}=\frac{5^{2010}+1+4}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)
Vì \(\frac{4}{5^{2011}+1}< \frac{4}{5^{2010}+1}\Rightarrow5A< 5B\Rightarrow A< B\)
Ta có:
A = \(\frac{5^{2010}+1}{5^{2011}+1}\)
\(\Rightarrow5A=\frac{5.\left(5^{2010}+1\right)}{5^{2011}+1}\)\(=\frac{5^{2011}+5}{5^{2011}+1}=1+\frac{4}{5^{2011}+1}\)
B=\(\frac{5^{2009}+1}{5^{2010}+1}\)
\(\Rightarrow5B=\frac{5.\left(5^{2009}+1\right)}{5^{2010}+1}=\frac{5^{2010}+5}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)
Ta thấy \(5^{2011}+1>5^{2010}+1\)
\(\Rightarrow\frac{4}{5^{2011}+1}< \frac{4}{5^{2010}+1}\)
\(\Rightarrow1+\frac{4}{5^{2011}+1}< 1+\frac{4}{5^{2010}+1}\)
Hay 5.A<5.B
Vậy A<B (đpcm)
\(5A=\frac{5^{2011}+5}{5^{2011}+1}=1+\frac{4}{5^{2011}+1}\)
\(5B=\frac{5^{2010}+5}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)
\(5B>5A\Rightarrow B>A\)
Ta có:
A = \(\frac{5^{2010}+1}{5^{2011}+1}\)
5A = \(\frac{5^{2011}+5}{5^{2011}+1}\) = \(\frac{5^{2011}+1+4}{5^{2011}+1}\) = 1 + \(\frac{4}{5^{2011}+1}\)
B = \(\frac{5^{2009}+1}{5^{2010}+1}\)
5B = \(\frac{5^{2010}+5}{5^{2010}+1}\) = \(\frac{5^{2010}+1+4}{5^{2010}+1}\) = 1 + \(\frac{4}{5^{2010}+1}\)
Vì 1 + \(\frac{4}{5^{2011}+1}\) < \(\frac{4}{5^{2010}+1}\) => 5A < 5B
Vì 5A < 5B => A < B
\(A=\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2010}\)
\(A=\frac{4064340600}{4066362660}+\frac{4064341605}{4066362660}+\frac{4070408792}{4066362660}\)
\(A=3,000000742\)
\(B=\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+....+\frac{1}{17}\)
\(B=1,939552553\)
vì đây là so sánh hai dòng phân số nên ta đổi ra thập phân nhé
do 3,000000742 > 1,939552553 và 3 > 1 Nên A > B nhé
đúng thì k nhé
chúc học giỏi !!!!
\(B=\frac{2009^{2010}-2}{2009^{2011}-2}< 1\)
\(\Rightarrow B=\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2010}-2+2011}{2009^{2011}-2+2011}=\frac{2009^{2010}+2009}{2009^{2011}+2009}\)\(=\frac{2009.\left(2009^{2009}+1\right)}{2009.\left(2009^{2010}+1\right)}=\frac{2009^{2009}+1}{2009^{2010}+1}\)
Suy ra : \(\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2009}+1}{2009^{2010}+1}\) hay \(B< A\)
Vậy \(A>B\)
\(B=\frac{2008+2009+2010}{2009+2010+2011}\)
\(=\frac{2008}{2009+2010+2011}+\frac{2009}{2009+2010+2011}+\frac{2010}{2009+2010+2011}\)
\(< \frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}=A\)
Ta có :
\(B=\frac{5^{2009}+1}{5^{2010}+1}=\frac{\left(5^{2009}+1\right).10}{\left(5^{2010}+1\right).10}=\frac{5^{2010}+10}{5^{2011}+10}\)
Ta thấy :
\(5^{2010}=5^{2010};1< 10\Rightarrow5^{2010}+1< 5^{2010}+10\)
\(5^{2011}=5^{2011};1< 10\Rightarrow5^{2011}+1< 5^{2011}+10\)
Suy ra : \(A< B\)
Vậy \(A< B\)
\(A< 1\)
\(A< \frac{5^{2010}+1}{5^{2011}+1}\)
\(A< \frac{5^{2010}+1+4}{5^{2011}+1+4}\)
\(A< \frac{5^{2010}+5}{5^{2011}+5}\)
\(A< \frac{5\left(5^{2009}+1\right)}{5\left(5^{2010}+1\right)}\)
\(A< \frac{5^{2009}+1}{5^{2010}+1}\)
\(A< B\)