a=7/10+7/10^2+7/10^3+...+7/10^2011
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Cho A= \(\frac{10^{2011+5}}{10^{2011}-2}\); B= \(\frac{10^{2011}}{10^{2011}-7}\). Hãy so sánh A và B
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\(A=\frac{10^{2011}+5}{10^{2011}-2}=\frac{10^{2011}-2+7}{10^{2011}-2}=1+\frac{7}{10^{2011}-2}\)
\(B=\frac{10^{2011}}{10^{2011}-7}=\frac{10^{2011}-7+7}{10^{2011}-7}=1+\frac{7}{10^{2011}-7}\)
Vì \(\frac{7}{10^{2011}-2}< \frac{7}{10^{2011}-7}\Rightarrow1+\frac{7}{10^{2011}-2}< 1+\frac{7}{10^{2011}-7}\Rightarrow A< B\)
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=2011^2011.(7^2-3.2^4-1)
=2011^2011.(49-3.16-1)
=2011^2011.(49-48-1)
=2011^2011.0
=0
Vote nhá
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\(2011^{2011}.\left(7^{10}:7^8-3.2^4-2^{2011}:2^{2011}\right)\)
\(=2011^{2011}.\left(7^{10-8}-3.2^4-2^{2011-2011}\right)\)
\(=2011^{2011}.\left(7^2-3.2^4-2^0\right)\)
\(=2011^{2011}.\left(49-3.16-1\right)\)
\(=2011^{2011}.\left(49-48-1\right)\)
\(=2011^{2011}.\left(1-1\right)\)
\(=2011^{2011}.0\)
\(=0\)
\(A=\dfrac{7}{10}+\dfrac{7}{10^2}+\dfrac{7}{10^3}+...+\dfrac{7}{10^{2011}}\)
\(\Rightarrow10A=7+\dfrac{7}{10}+\dfrac{7}{10^2}+...+\dfrac{7}{10^{2010}}\)
\(\Rightarrow10A-A=7+\dfrac{7}{10}+\dfrac{7}{10^2}+...+\dfrac{7}{10^{2010}}-\left(\dfrac{7}{10}+\dfrac{7}{10^2}+\dfrac{7}{10^3}+...+\dfrac{7}{10^{2011}}\right)\)
\(\Rightarrow9A=7-\dfrac{7}{10^{2011}}\)
\(\Rightarrow A=\dfrac{7}{9}.\left(1-\dfrac{1}{10^{2011}}\right)\)