Cho
\(A=\frac{2}{60\cdot63}+\frac{2}{61\cdot64}+...+\frac{2}{117\cdot120}+2011\)
\(B=\frac{5}{40\cdot44}+\frac{5}{44\cdot48}+...+\frac{5}{76\cdot80}+\frac{2}{2011}\)
Hãy so sánh A và B
K
Khách
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13 tháng 4 2019
Bài 2:
Ta có: A=\(2\left(\frac{1}{60.63}+\frac{1}{63.66}+\frac{1}{66.69}+...+\frac{1}{117.120}+\frac{1}{2011}\right)\)
\(=2\left(\frac{3}{60.63}+\frac{3}{63.66}+....+\frac{3}{117.120}+\frac{3}{2011}\right).\frac{1}{3}\)
\(=2\left(\frac{1}{60}-\frac{1}{63}+\frac{1}{63}-\frac{1}{66}+...+\frac{1}{117}-\frac{1}{120}+\frac{3}{2011}\right).\frac{1}{3}\)
\(=2\left(\frac{1}{60}-\frac{1}{120}+\frac{3}{2011}\right).\frac{1}{3}\)\(=\frac{2}{3}.\left(\frac{1}{120}+\frac{3}{2011}\right)=\frac{2}{3}.\frac{1}{120}+\frac{3}{2011}.\frac{2}{3}\)
\(=\frac{1}{180}+\frac{2}{2011}\)
B=\(5\left(\frac{1}{40.44}+\frac{1}{44.48}+...+\frac{1}{76.80}\right)+\frac{5}{2011}\)
\(=\frac{5}{4}\left(\frac{1}{40}-\frac{1}{44}+\frac{1}{44}-\frac{1}{48}+...+\frac{1}{76}-\frac{1}{80}\right)+\frac{5}{2011}\)
\(=\frac{5}{4}\left(\frac{1}{40}-\frac{1}{80}\right)+\frac{5}{2011}=\frac{5}{4}.\frac{1}{80}+\frac{5}{2011}\)\(=\frac{1}{64}+\frac{5}{2011}\)
Xét: \(\frac{1}{180}< \frac{1}{64};\frac{2}{2011}< \frac{5}{2011}\)
\(\Rightarrow\frac{1}{180}+\frac{2}{2011}< \frac{1}{64}+\frac{5}{2011}\)
\(\Leftrightarrow A< B\)
Vậy: A<B
9 tháng 5 2019
Bài 3: Ta có:
C=222...22000...00777....7
( có 2011 c/s 2; 2011 c/s 0; 2011 c/s 7)
\(\Rightarrow\) Tổng các c/s của C là:
2011.2+2011.0+2011.7=18099=9.2011 \(⋮9\)
\(\Rightarrow C⋮9\)
Vậy C có ít nhất 3 ước: 1;C và C.
Từ đó suy ra C là hợp số.
Vậy C là hợp số.
17 tháng 2 2017
Ta có: \(A=\frac{2}{60.63}+\frac{2}{63.66}+...+\frac{2}{117.120}+\frac{2}{2003}\)
\(\Rightarrow A=\frac{2}{3}\left(\frac{3}{60.63}+\frac{3}{63.66}+...+\frac{3}{117.120}\right)+\frac{2}{2003}\)
\(\Rightarrow A=\frac{2}{3}\left(\frac{1}{60}-\frac{1}{63}+\frac{1}{63}-\frac{1}{66}+...+\frac{1}{117}-\frac{1}{120}\right)+\frac{2}{2003}\)
\(\Rightarrow A=\frac{2}{3}\left(\frac{1}{60}-\frac{1}{120}\right)+\frac{2}{2003}\)
\(\Rightarrow A=\frac{2}{3}.\frac{1}{120}+\frac{2}{2003}\)
\(\Rightarrow A=\frac{1}{180}+\frac{2}{2003}\)
\(B=\frac{5}{40.44}+\frac{5}{44.48}+...+\frac{5}{76.80}+\frac{5}{2003}\)
\(\Rightarrow B=\frac{5}{4}\left(\frac{4}{40.44}+\frac{4}{44.48}+...+\frac{4}{76.80}\right)+\frac{5}{2003}\)
\(\Rightarrow B=\frac{5}{4}\left(\frac{1}{40}-\frac{1}{44}+\frac{1}{44}-\frac{1}{48}+...+\frac{1}{76}-\frac{1}{80}\right)+\frac{5}{2003}\)
\(\Rightarrow B=\frac{5}{4}\left(\frac{1}{40}-\frac{1}{80}\right)+\frac{5}{2003}\)
\(\Rightarrow B=\frac{5}{4}.\frac{1}{80}+\frac{5}{2003}\)
\(\Rightarrow B=\frac{1}{64}+\frac{5}{2003}\)
Vì \(\left\{\begin{matrix}\frac{1}{64}>\frac{1}{180}\\\frac{5}{2003}>\frac{2}{2003}\end{matrix}\right.\Rightarrow\frac{1}{64}+\frac{5}{2003}>\frac{1}{180}+\frac{2}{2003}\Rightarrow B>A\)
Vậy A < B
NN
Cho A= \(\frac{10^{2011+5}}{10^{2011}-2}\); B= \(\frac{10^{2011}}{10^{2011}-7}\). Hãy so sánh A và B
1
S
13 tháng 5 2017
\(A=\frac{10^{2011}+5}{10^{2011}-2}=\frac{10^{2011}-2+7}{10^{2011}-2}=1+\frac{7}{10^{2011}-2}\)
\(B=\frac{10^{2011}}{10^{2011}-7}=\frac{10^{2011}-7+7}{10^{2011}-7}=1+\frac{7}{10^{2011}-7}\)
Vì \(\frac{7}{10^{2011}-2}< \frac{7}{10^{2011}-7}\Rightarrow1+\frac{7}{10^{2011}-2}< 1+\frac{7}{10^{2011}-7}\Rightarrow A< B\)
18 tháng 9 2018
B=5(1/12−1/21+1/21−1/30)−5(1/24−1/34+1/34−1/44+1/44−1/54+1/54−1/64)
B=5(1/12−1/21+1/21−1/30+1/24−1/34+1/34−1/44+1/44−1/54+1/54−1/64 )
B=5(1/12−1/64)=5.13/192=65/192
T
1 tháng 7 2021
\(\text{A = }\frac{\text{-1}}{\text{2011}}-\frac{\text{3}}{\text{11}^2}-\frac{\text{5}}{\text{11}^2.\text{11}}-\frac{\text{7}}{\text{11}^2.\text{11}^2}=\text{ }\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}\right)\)
\(\text{B = }\text{ }\frac{\text{-1}}{\text{2011}}-\frac{7}{\text{11}^2}-\frac{5}{\text{11}^2.\text{11}}-\frac{3}{\text{11}^2.\text{11}^2}=\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\right)\)
\(\text{Vì }3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}< 7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\)
\(\Rightarrow\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}\right)>\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\right)\)
=> A > B
Vậy A > B
T
1 tháng 7 2021
\(\text{A = }\frac{\text{-1}}{\text{2011}}-\frac{\text{3}}{\text{11}^2}-\frac{\text{5}}{\text{11}^2.\text{11}}-\frac{\text{7}}{\text{11}^2.\text{11}^2}=\text{ }\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}\right)\)
\(\text{B = }\frac{\text{-1}}{\text{2011}}-\frac{7}{\text{11}^2}-\frac{5}{\text{11}^2.\text{11}}-\frac{3}{\text{11}^2.\text{11}^2}=\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\right)\)
\(\text{Vì }3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}< 7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\)
\(\Rightarrow\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}\right)>\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\right)\)
=> A > B
Vậy A > B
T
1 tháng 7 2021
\(\text{A = }\frac{\text{-1}}{\text{2011}}-\frac{\text{3}}{\text{11}^2}-\frac{\text{5}}{\text{11}^2.\text{11}}-\frac{\text{7}}{\text{11}^2.\text{11}^2}=\text{ }\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}\right)\)
\(\text{B = }\frac{\text{-1}}{\text{2011}}-\frac{7}{\text{11}^2}-\frac{5}{\text{11}^2.\text{11}}-\frac{3}{\text{11}^2.\text{11}^2}=\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\right)\)
\(\text{Vì }3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}< 7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\)
\(\Rightarrow\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(3-\frac{\text{5}}{\text{11}}-\frac{\text{7}}{\text{11}^2}\right)>\frac{\text{-1}}{\text{2011}}-\frac{\text{1}}{\text{11}^2}.\left(7-\frac{5}{\text{11}}-\frac{3}{\text{11}^2}\right)\)
=> A > B
Vậy A > B