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\(P=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)...\left(1-\frac{1}{100^2}\right)\)
\(=\frac{2^2-2}{2^2}.\frac{3^2-1}{3^2}...\frac{100^2-1}{100^2}\)
\(=\frac{1.3}{2^2}.\frac{2.4}{3^2}...\frac{99.101}{100^2}\)
\(=\frac{1.2...99}{2.3...100}.\frac{3.4...101}{2.3...100}\)
\(=\frac{1}{100}.\frac{101}{2}\)
\(=\frac{101}{200}\)
\(P=\left(1-\frac{1}{2^2}\right).\left(1-\frac{1}{3^2}\right)......\left(1-\frac{1}{100^2}\right)\)
\(\Rightarrow P=\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}......\frac{100^2-1}{100^2}\)
\(\Rightarrow P=\frac{3}{2^2}.\frac{8}{3^2}...............\frac{9999}{100^2}\)
\(\Rightarrow P=\frac{1.3}{2^2}.\frac{2.4}{3^2}............\frac{99.101}{100^2}\)
\(\Rightarrow P=\frac{\left(1.2............99\right).\left(3.4............101\right)}{\left(2.3..............100\right).\left(2.3...........100\right)}\)
\(\Rightarrow P=\frac{1.101}{100.2}=\frac{101}{200}\)
Vậy \(P=\frac{101}{200}\)
Chúc bn học tốt
a) \(15+2\left|x\right|=-3\\ \\ < =>2\left|x\right|=15-\left(-3\right)\\ < =>2\left|x\right|=18\\ =>\left|x\right|=\frac{18}{2}=9\\ =>x=9hoặcx=-9\)
b) \(\left|x-2\right|=7\\ < =>x-2=7hoặcx-2=-7\\ =>x=9hoặcx=-5\)
c) \(100-4.x^2=224\\ < =>4.x^2=100-224=-124\\ < =>x^2=-\frac{124}{4}=-31\\ Mà:x^2\ge0\\ =>xkhôngcógiátrịnàothoảmãn\)
d)\(2x-\frac{9}{240}=\frac{39}{80}\\ < =>2x-\frac{3}{80}=\frac{39}{80}\\ =>2x=\frac{39}{80}+\frac{3}{80}=\frac{21}{40}\\ =>x=\frac{\frac{21}{40}}{2}=\frac{21}{80}\)
A=33. \(\left(1-\frac{2}{3}\right)\left(1-\frac{2}{5}\right)...\left(1-\frac{2}{99}\right)\)
A=33.\(\frac{1}{3}.\frac{3}{5}....\frac{97}{99}\)
A=33.\(\frac{1}{99}\)
A=\(\frac{33}{99}=\frac{1}{3}\)
\(a=\frac{2011\cdot2010+2000}{2010\cdot2012-10}=\frac{2011\cdot2010+2000}{2010\cdot2011+2010-10}=\frac{2011\cdot2010+2000}{2010\cdot2011+2000}=1\)
\(b=\frac{2013}{2012}>1\) (vì 2013 > 2012)
\(\Rightarrow\) a < b
\(\frac{1}{1\times2}+\frac{1}{2\times3}+.......+\frac{1}{2015\times2016}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.........+\frac{1}{2015}-\frac{1}{2016}\)
\(=1-\frac{1}{2016}\)
\(=\frac{2015}{2016}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2015}-\frac{1}{2016}\)
\(=\frac{1}{1}-\frac{1}{2016}\)
\(=\frac{2015}{2016}\)
ta có -25/4=-175/28
21/7=84/28
=>25/7< -174/28< -173/28<....<83/24<82428
=> ... là -174/28<173/28...<83/28
gọi số cần tìm là x=y/13
để x ngyên =>13 thuộc ư(y)
=>y ={-13;13;26;-26;39;-39...}
vì -34/13 < y/31
=> y={-13;-26}
mà x nhỏ nhất => y nhỏ nhất
=>y =26
=> phân số cần tìm là -26/13=-2
\(Q=\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{19}+\frac{1}{19}-\frac{1}{31}+\frac{1}{31}-\frac{1}{101}+\frac{1}{101}-\frac{1}{200}\)
\(Q=\frac{1}{5}-\frac{1}{200}=\frac{39}{200}\)
\(Q=\frac{3}{5.8}+\frac{11}{8.19}+\frac{12}{19.31}+\frac{70}{31.101}+\frac{99}{101.200}\)
\(=\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{19}+\frac{1}{19}-\frac{1}{31}+\frac{1}{31}-\frac{1}{101}+\frac{1}{101}-\frac{1}{200}\)
\(=\frac{1}{5}-\frac{1}{200}\)
\(=\frac{39}{200}\)