Tính nhanh:
\(\frac{2003\cdot14+1988+2005\cdot2002}{2002+1001\cdot1006+5060\cdot2002}\)
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(2002+1)x14+1988+2001x2002
2002x(1+503+504)
2002x14+14+1988+2001x2002
2002x1008
2002x14+2002+2001x2002
2002x1008
2002x(14+2001+1)
2002x1008
2002x2016
2002x1008
2
1
1/2003.2002 - 1/2002.2001 - ... - 1/3.2 - 1/2.1
= 1/2003.2002 - (1/2002.2001 + ... + 1/3.2 + 1/2.1)
= 1/2002.2003 - (1/1.2 + 1/2.3 + ... + 1/2001.2002)
= 1/2002.2003 - (1 - 1/2 + 1/2 - 1/3 + ... + 1/2001 - 1/2002)
= 1/2002.2003 - (1 - 1/2002)
= 1/2002.2003 - 2001/2002
= 1/2002.2003 - 2001.2003/2002.2003
= 1-2001.2003/2002.2003
P=\(\frac{\left(2002+1\right)\times14+1988+2001\times2002}{2002\times\left(1+503+504\right)}\)
\(=\frac{2002\times14+2002+2001\times2002}{2002\times1008}\)
\(=\frac{2002\times\left(14+1+2001\right)}{2002\times1008}=\frac{2016}{1008}=2\)
\(P=\frac{\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}}{\frac{5}{2003}+\frac{5}{2004}-\frac{5}{2005}}-\frac{\frac{2}{2002}+\frac{2}{2003}-\frac{2}{2004}}{\frac{3}{2002}+\frac{3}{2003}-\frac{3}{2004}}\)
\(=\frac{\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}}{5\left(\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}\right)}-\frac{2\left(\frac{1}{2002}+\frac{1}{2003}-\frac{1}{2004}\right)}{3\left(\frac{1}{2002}+\frac{1}{2003}-\frac{1}{2004}\right)}\)
\(=\frac{1}{5}-\frac{2}{3}=-\frac{7}{15}\)
Ta có:
\(P=\frac{\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}}{\frac{5}{2003}+\frac{5}{2004}-\frac{5}{2005}}-\frac{\frac{2}{2002}+\frac{2}{2003}-\frac{2}{2004}}{\frac{3}{2002}+\frac{3}{2003}-\frac{3}{2004}}\)
\(P=\frac{1}{5}\cdot\left(\frac{\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}}{\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}}\right)-\frac{2}{3}\cdot\left(\frac{\frac{1}{2002}+\frac{1}{2003}-\frac{1}{2004}}{\frac{1}{2002}+\frac{1}{2003}-\frac{1}{2004}}\right)\)
\(P=\frac{1}{5}-\frac{2}{3}=-\frac{7}{15}\)
Ta có: 2003x14 = 2002x14 + 14
=> 2003x14 + 1988 + 2001x2002 = 2002x14 + 2002 + 2001x2002 = 2016x2002
Lại có: 2002+2002x503+504x2002 = 1008x2002
=> 2003x14+1988+2001x2002 2016x2002
-------------------------------- = --------------------- = 2
2002+2002x503+504x2002 1008x2002
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