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\(A=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)...\left(\frac{1}{121}-1\right)\)
\(-A=\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)...\left(1-\frac{1}{121}\right)\)
\(-A=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{120}{121}\)
\(-A=\frac{1\cdot3\cdot2\cdot4\cdot3\cdot5\cdot...\cdot10\cdot12}{2\cdot2\cdot3\cdot3\cdot4\cdot4\cdot...\cdot11\cdot11}\)
\(-A=\frac{\left(1\cdot2\cdot3\cdot...\cdot10\right)\left(3\cdot4\cdot5\cdot...\cdot12\right)}{\left(2\cdot3\cdot4\cdot...\cdot11\right)\left(2\cdot3\cdot4\cdot...\cdot11\right)}\)
\(-A=\frac{1\cdot12}{11\cdot2}=\frac{6}{11}\)
\(A=-\frac{6}{11}\)
\(B=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{37\cdot38}\)
\(B=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{37}-\frac{1}{38}\)
\(B=1-\frac{1}{38}=\frac{37}{38}\)
\(M=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+..+\frac{1}{37.38}-\frac{1}{38.39}\)
\(=\frac{1}{1.2}-\frac{1}{38.39}=\frac{1}{1}-\frac{1}{2}-\frac{1}{38}+\frac{1}{39}=\frac{370}{741}\)
A = 1 - 1/2 + 1/3 - 1/4 + ..... + 1/19 - 1/20
= 1 + 1/2 + 1/3 + 1/4 +............+ 1/19 + 1/20 - 2(1/2 + 1/4 + 1/6 +...........+ 1/20)
= 1 + 1/2 + 1/3 + 1/4 +............+ 1/19 + 1/20 - ( 1 + 1/2 + 1/3 + .............. + 1/10)
= 1/11 + 1/12 + ...... + 1/20
=> A / B = 1
A=1/1x2 +1/2x3 +... +1/18x19 + 1/19x20
Nhận xét 1/1x2 = 1/1 -1/2 ; 1/2x3=1/2-1/3; ... ;1/18x19=1/18-1/19 ; 1/19x20=1/19-1/20
ta có A=1/1 - 1/2 + +1/2 -1/3+1/3- +1/18-1/19+1/19-1/20
A=1/1 - 1/20
A=20/20 - 1/20
A=(20-1)/20
A=19/20
Vậy A=19/20
A =\(\frac{1}{1.2}\)+\(\frac{1}{2.3}\)+\(\frac{1}{3.4}\)+ ...+\(\frac{1}{18.19}\)+\(\frac{1}{19.20}\)
A = 1 - \(\frac{1}{2}\)+\(\frac{1}{2}\)- \(\frac{1}{3}\)+\(\frac{1}{3}\)- \(\frac{1}{4}\)+....+\(\frac{1}{18}\)- \(\frac{1}{19}\)+ \(\frac{1}{19}\)- \(\frac{1}{20}\)
A = 1 - \(\frac{1}{20}\)( Vì đã triệt tiêu )
A = \(\frac{19}{20}\)
bài này mình đã học rồi nhưng nó nâng cao , bạn nhé !
Chúc bạn học tốt ! 
Ta có \(M=\frac{1}{125\cdot2^3}\cdot\frac{1}{125\cdot3^3}\cdot\frac{1}{125\cdot4^3}\cdot...\cdot\frac{1}{125\cdot20^3}\)
\(\Rightarrow M=0\)