(1)/(2)-(43)/(101)+(-(1)/(3))-(1)/(6)
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1/2 -43/101+(-1/3)-1/6
= -43/101+(-1/3)-1/6+1/2
=-43/101+0
=-43/101
\(M=1-2+3-4+5-6+......+99-100+101\)
\(=\left(1+3+5+.......+99+101\right)-\left(2+4+6+.......+100\right)\)
\(=\left(1+2+3+4+........+100+101\right)-2.\left(2+4+6+......+100\right)\)
\(=\left(1+2+3+4+.......+100+101\right)-\left(1+2+3+.......+50\right)\)
\(=51+52+53+.......+101=\frac{\left(101+51\right).51}{2}=\frac{152.51}{2}=3876\)
Giải:
\(\dfrac{1}{2}-\dfrac{43}{100}+\left(-\dfrac{1}{3}\right)-\dfrac{1}{6}\)
\(=-\dfrac{43}{100}+\dfrac{1}{2}+\left(-\dfrac{1}{3}\right)-\dfrac{1}{6}\)
\(=-\dfrac{43}{100}+\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}\)
\(=-\dfrac{43}{100}+\left(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}\right)\)
\(=-\dfrac{43}{100}+\left(\dfrac{3}{6}-\dfrac{2}{6}-\dfrac{1}{6}\right)\)
\(=-\dfrac{43}{100}+0=-\dfrac{43}{100}\)
Vậy ...
\(\frac{1}{2}-\frac{43}{101}+\frac{-1}{3}-\frac{1}{6}\)
\(=\left(\frac{1}{2}-\frac{1}{3}-\frac{1}{6}\right)+\left(-\frac{43}{101}\right)\)
\(=\frac{3-2-1}{6}+\frac{-43}{101}\)
\(=0+\frac{-43}{101}\)
\(=-\frac{43}{101}\)
Đặt S = \(\frac{1}{2}+\frac{1}{2^5}+\frac{1}{2^9}+...+\frac{1}{2^{101}}\)
=> 24S = 16S = \(2^3+\frac{1}{2}+\frac{1}{2^5}+...+\frac{1}{2^{97}}\)
=> 16S - S = \(2^3+\frac{1}{2}+\frac{1}{2^5}+...+\frac{1}{2^{97}}-\left(\frac{1}{2}+\frac{1}{2^5}+\frac{1}{2^9}+...+\frac{1}{2^{101}}\right)\)
=> 15S = \(2^3-\frac{1}{2^{101}}\)
=> S = \(\frac{2^3-\frac{1}{2^{101}}}{15}\)
Khi đó A = \(\frac{2^3-\frac{1}{2^{101}}}{15}:\left(2^3-\frac{1}{2^{101}}\right)=\frac{1}{15}\)
\(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}=1-\dfrac{1}{101}=\dfrac{100}{101}\)
\(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{99\cdot101}\\ =1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\\ =1-\dfrac{1}{101}=\dfrac{100}{101}\)
\(\frac{101+100+99+98+...+3+2+1}{101-100+99-98+...+3-2+1}\)
\(=\frac{\frac{101.102}{2}}{51}\)
\(=101\)
\(\dfrac{1}{2}-\dfrac{43}{101}+\dfrac{-1}{3}-\dfrac{1}{6}\\ =\left(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}\right)-\dfrac{43}{101}=0-\dfrac{43}{101}=-\dfrac{43}{101}\)