Gỉa sử p và q là các số nguyên sao cho
\(\dfrac{p}{q}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4} +...-\dfrac{1}{1334}+\dfrac{1}{1335}\)
CMR: p \(⋮\)2003
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Gỉa sử p và q là các số nguyên sao cho:
\(\dfrac{p}{q}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...-\dfrac{1}{1334}+\dfrac{1}{1335}\)
Chứng minh rằng p\(⋮\) 2003
Giả sử m và n là các số nguyên sao cho:\(\dfrac{m}{n}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...-\dfrac{1}{1334}+\dfrac{1}{1335}\) .Chứng minh rằng m chia hết cho 2003
Tính giá trị biểu thức:
Ddfrac{dfrac{1}{2003}+dfrac{1}{2004}+dfrac{1}{2005}}{dfrac{5}{2003}+dfrac{5}{2004}+dfrac{5}{2005}}-dfrac{dfrac{2}{2002}+dfrac{2}{2003}+dfrac{2}{2004}}{dfrac{2}{2002}+dfrac{3}{2003}+dfrac{3}{2004}}
Hdfrac{dfrac{1}{2}+dfrac{1}{3}+dfrac{1}{4}+...+dfrac{1}{2012}}{dfrac{2011}{1}+dfrac{2010}{2}+...+dfrac{1}{2011}}
Idfrac{dfrac{1}{2}+dfrac{1}{3}+dfrac{1}{4}+...+dfrac{1}{2012}}{dfrac{2012}{2}+dfrac{2012}{3}+...+dfrac{2012}{2011}}
Help me!
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Tính giá trị biểu thức:
\(D=\dfrac{\dfrac{1}{2003}+\dfrac{1}{2004}+\dfrac{1}{2005}}{\dfrac{5}{2003}+\dfrac{5}{2004}+\dfrac{5}{2005}}-\dfrac{\dfrac{2}{2002}+\dfrac{2}{2003}+\dfrac{2}{2004}}{\dfrac{2}{2002}+\dfrac{3}{2003}+\dfrac{3}{2004}}\)
\(H=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{\dfrac{2011}{1}+\dfrac{2010}{2}+...+\dfrac{1}{2011}}\)
\(I=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{\dfrac{2012}{2}+\dfrac{2012}{3}+...+\dfrac{2012}{2011}}\)
Help me!
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\(D=\dfrac{\dfrac{1}{2003}+\dfrac{1}{2004}+\dfrac{1}{2005}}{\dfrac{5}{2003}+\dfrac{5}{2004}+\dfrac{5}{2005}}-\dfrac{\dfrac{2}{2002}+\dfrac{2}{2003}+\dfrac{2}{2004}}{\dfrac{3}{2002}+\dfrac{3}{2003}+\dfrac{3}{2004}}\)
\(D=\dfrac{1.\left(\dfrac{1}{2003}+\dfrac{1}{2004}+\dfrac{1}{2005}\right)}{5.\left(\dfrac{1}{2003}+\dfrac{1}{2004}+\dfrac{1}{2005}\right)}-\dfrac{2.\left(\dfrac{1}{2002}+\dfrac{1}{2003}+\dfrac{1}{2004}\right)}{3\left(\dfrac{1}{2002}+\dfrac{1}{2003}+\dfrac{1}{2004}\right)}\)
\(D=\dfrac{1}{5}-\dfrac{2}{3}\)
\(D=-\dfrac{7}{15}\)
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Tính giá trị của các biểu thức sau1) A1+2+2^2+...+2^{2015}2) Bleft(dfrac{1}{4}-1right)cdotleft(dfrac{1}{9}-1right)cdotleft(dfrac{1}{16}-1right)cdotcdotcdotcdotcdotleft(dfrac{1}{400}-1right)3) Cleft(dfrac{1}{4cdot9}+dfrac{1}{9cdot14}+dfrac{1}{14cdot19}+...+dfrac{1}{44cdot49}right)cdotdfrac{1-3-5-7-...-49}{89}4) Ddfrac{2^{12}cdot3^5-4^6cdot9^2}{left(2^2cdot3right)^6+8^4cdot3^5}-dfrac{5^{10}cdot7^3-25^5cdot49^2}{left(125cdot7right)^3+5^9cdot14^3}5) Edfrac{dfrac{1}{2003}+dfrac{1}{2004}-dfrac{1}{2005...
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Tính giá trị của các biểu thức sau
1) \(A=1+2+2^2+...+2^{2015}\)
2) \(B=\left(\dfrac{1}{4}-1\right)\cdot\left(\dfrac{1}{9}-1\right)\cdot\left(\dfrac{1}{16}-1\right)\cdot\cdot\cdot\cdot\cdot\left(\dfrac{1}{400}-1\right)\)
3) \(C=\left(\dfrac{1}{4\cdot9}+\dfrac{1}{9\cdot14}+\dfrac{1}{14\cdot19}+...+\dfrac{1}{44\cdot49}\right)\cdot\dfrac{1-3-5-7-...-49}{89}\)
4) \(D=\dfrac{2^{12}\cdot3^5-4^6\cdot9^2}{\left(2^2\cdot3\right)^6+8^4\cdot3^5}-\dfrac{5^{10}\cdot7^3-25^5\cdot49^2}{\left(125\cdot7\right)^3+5^9\cdot14^3}\)
5) \(E=\dfrac{\dfrac{1}{2003}+\dfrac{1}{2004}-\dfrac{1}{2005}}{\dfrac{5}{2003}+\dfrac{5}{2004}-\dfrac{5}{2005}}-\dfrac{\dfrac{2}{2002}+\dfrac{2}{2003}-\dfrac{2}{2004}}{\dfrac{3}{2002}+\dfrac{3}{2003}-\dfrac{3}{2004}}\)
6) Cho 13+23+...+103=3025
Tính S= 23+43+63+...+203
Bài 1: Thực hiện phép tính:
a) ( 2- dfrac{3}{2}). ( 2- dfrac{4}{3}). (2- dfrac{5}{4}). ( 2- dfrac{6}{5})
b) dfrac{1}{2002}+dfrac{2003.2001}{2002}-2003
c)( dfrac{2003}{2004}+dfrac{2004}{2003}):dfrac{8028045}{8028024}
d) 4+ dfrac{1}{1+dfrac{1}{1+dfrac{2}{1+dfrac{3}{4}}}}
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Bài 1: Thực hiện phép tính:
a) ( 2- \(\dfrac{3}{2}\)). ( 2- \(\dfrac{4}{3}\)). (2- \(\dfrac{5}{4}\)). ( 2- \(\dfrac{6}{5}\))
b) \(\dfrac{1}{2002}+\dfrac{2003.2001}{2002}-2003\)
c)( \(\dfrac{2003}{2004}+\dfrac{2004}{2003}\)):\(\dfrac{8028045}{8028024}\)
d) 4+ \(\dfrac{1}{1+\dfrac{1}{1+\dfrac{2}{1+\dfrac{3}{4}}}}\)
a, Cho Adfrac{1}{11}+dfrac{1}{12}+dfrac{1}{13}+...+dfrac{1}{70}. CMR: dfrac{4}{3} A dfrac{5}{2}
b, Cho Adfrac{1}{2}-dfrac{1}{3}+dfrac{1}{4}-dfrac{1}{5}+...+dfrac{1}{98}-dfrac{1}{99}.CMR: 0,2 A 0,4
c, Cho Adfrac{1}{2}.dfrac{3}{4}.dfrac{5}{6}...dfrac{99}{100}. CMR: dfrac{1}{15} A dfrac{1}{10}
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a, Cho A=\(\dfrac{1}{11}+\dfrac{1}{12}+\dfrac{1}{13}+...+\dfrac{1}{70}\). CMR: \(\dfrac{4}{3}< A< \dfrac{5}{2}\)
b, Cho \(A=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{98}-\dfrac{1}{99}\).CMR: \(0,2< A< 0,4\)
c, Cho \(A=\dfrac{1}{2}.\dfrac{3}{4}.\dfrac{5}{6}...\dfrac{99}{100}\). CMR: \(\dfrac{1}{15}< A< \dfrac{1}{10}\)
1, P dfrac{dfrac{1}{2003}+dfrac{1}{2004}-dfrac{1}{2005}}{dfrac{5}{2003}+dfrac{5}{2004}-dfrac{5}{2005}} - dfrac{dfrac{2}{2002}+dfrac{2}{2003}-dfrac{2}{2004}}{dfrac{3}{2002}+dfrac{3}{2003}-dfrac{2}{2004}}
2, Q ( dfrac{1,5+1-0,75}{2,5+dfrac{5}{3}-1,25} + dfrac{0,375-0,3+dfrac{3}{11}+dfrac{3}{12}}{-0,625+0,5-dfrac{5}{11}-dfrac{5}{12}} ) : dfrac{1980}{3758} + 155
3, A 1.3 + 2.4 + 3.5 +....+ 97.99 + 98.100
4, B 1.2.3 + 2.3.4. +...+ 48.49.50
5, C dfrac{1}{1.2.3.4} + dfrac{1}{2.3.4.5} +...+ df...
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1, P = \(\dfrac{\dfrac{1}{2003}+\dfrac{1}{2004}-\dfrac{1}{2005}}{\dfrac{5}{2003}+\dfrac{5}{2004}-\dfrac{5}{2005}}\) - \(\dfrac{\dfrac{2}{2002}+\dfrac{2}{2003}-\dfrac{2}{2004}}{\dfrac{3}{2002}+\dfrac{3}{2003}-\dfrac{2}{2004}}\)
2, Q = ( \(\dfrac{1,5+1-0,75}{2,5+\dfrac{5}{3}-1,25}\) + \(\dfrac{0,375-0,3+\dfrac{3}{11}+\dfrac{3}{12}}{-0,625+0,5-\dfrac{5}{11}-\dfrac{5}{12}}\) ) : \(\dfrac{1980}{3758}\) + 155
3, A = 1.3 + 2.4 + 3.5 +....+ 97.99 + 98.100
4, B = 1.2.3 + 2.3.4. +...+ 48.49.50
5, C = \(\dfrac{1}{1.2.3.4}\) + \(\dfrac{1}{2.3.4.5}\) +...+ \(\dfrac{1}{27.28.29.30}\)
6, D = 1 + \(2^2\) + \(2^4\) + \(2^6\) + .... +\(2^{200}\)
7, E = \(\dfrac{1}{3.5}\)+ \(\dfrac{5}{5.7}\) +...+ \(\dfrac{1}{97.99}\)
6:
\(4D=2^2+2^4+...+2^{202}\)
=>3D=2^202-1
hay \(D=\dfrac{2^{202}-1}{3}\)
7: \(=\dfrac{1}{2}\left(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{97\cdot99}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{32}{99}=\dfrac{16}{99}\)
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tìm số tự nhiên thỏa mãn điều kiện
2cdot2^2+3cdot2^3+4cdot2^4+........+ncdot2^n2^{n+11}
rút gọn : Aleft(dfrac{2}{5}-dfrac{5}{2}+dfrac{1}{10}right):left(dfrac{5}{2}-dfrac{2}{3}+dfrac{1}{12}right)
tính:Bdfrac{dfrac{1}{2}+dfrac{1}{3}+dfrac{1}{4}+......+dfrac{1}{2017}}{dfrac{2016}{1}+dfrac{2003}{2}+dfrac{2002}{3}+.......+dfrac{1}{2016}}
CMR :5a+2b⋮13Leftrightarrow9a+b⋮13left(a,bin Zright)
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tìm số tự nhiên thỏa mãn điều kiện
\(2\cdot2^2+3\cdot2^3+4\cdot2^4+........+n\cdot2^n=2^{n+11}\)
rút gọn : \(A=\left(\dfrac{2}{5}-\dfrac{5}{2}+\dfrac{1}{10}\right):\left(\dfrac{5}{2}-\dfrac{2}{3}+\dfrac{1}{12}\right)\)
tính:\(B=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+......+\dfrac{1}{2017}}{\dfrac{2016}{1}+\dfrac{2003}{2}+\dfrac{2002}{3}+.......+\dfrac{1}{2016}}\)
CMR :\(5a+2b⋮13\Leftrightarrow9a+b⋮13\left(a,b\in Z\right)\)
8) Adfrac{9}{10}-dfrac{1}{90}-dfrac{1}{72}-dfrac{1}{56}-dfrac{1}{42}-dfrac{1}{30}-dfrac{1}{20}-dfrac{1}{12}-dfrac{1}{6}-dfrac{1}{2}
9) Bdfrac{1}{3}+dfrac{1}{3^2}+dfrac{1}{3^4}+...+dfrac{1}{3^{2014}}+dfrac{1}{3^{2015}}
10) Pdfrac{dfrac{1}{2}+dfrac{1}{3}+dfrac{1}{4}+...+dfrac{1}{2005}}{dfrac{2004}{1}+dfrac{2003}{2}+dfrac{2002}{3}+...+dfrac{1}{2004}}
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8) \(A=\dfrac{9}{10}-\dfrac{1}{90}-\dfrac{1}{72}-\dfrac{1}{56}-\dfrac{1}{42}-\dfrac{1}{30}-\dfrac{1}{20}-\dfrac{1}{12}-\dfrac{1}{6}-\dfrac{1}{2}\)
9) \(B=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{2014}}+\dfrac{1}{3^{2015}}\)
10) \(P=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2005}}{\dfrac{2004}{1}+\dfrac{2003}{2}+\dfrac{2002}{3}+...+\dfrac{1}{2004}}\)
8,A=\(\dfrac{9}{10}-\left(\dfrac{1}{10\times9}+\dfrac{1}{9\times8}+\dfrac{1}{8\times7}+...+\dfrac{1}{2\times1}\right)\)
=\(\dfrac{9}{10}-\left(\dfrac{1}{10}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{8}+...+\dfrac{1}{2}-1\right)\)
=\(\dfrac{9}{10}-\left(\dfrac{1}{10}-1\right)\)
=\(\dfrac{9}{10}-\dfrac{\left(-9\right)}{10}\)
=\(\dfrac{9}{5}\)
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