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đặt S=1.2.3+2.3.4+....+47.48.49
4S=1.2.3.(4-0)+2.3.4.(5-1)+...+47.48.49.(50-46)
4S=1.2.3.4-1.2.3+2.3.4.5-1.2.3.4+....+47.48.49.50-46.47.48.49
4S=47.48.49.50-1.2.3
S=(47.48.49.50-1.2.3):4
2Q=\(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+.........+\frac{1}{9.10}-\frac{1}{10.11}\)
2Q=\(\frac{1}{1.2}-\frac{1}{10.11}\)
2Q=\(\frac{1}{2}-\frac{1}{110}\)
2Q=\(\frac{55}{110}-\frac{1}{110}\)
2Q=\(\frac{54}{110}\)
Q=\(\frac{54}{110}:2\)
Q=\(\frac{27}{110}\)
A = 5/20.22 + 5/22.24+...+5/79.81
A = 5/2 . (2/20.22 + 2/22.24 + ... + 2/79.81)
A = 5/2 . (1/20 - 1/22 + 1/22 - 1/24 + ... + 1/79 - 1/81)
A = 5/2 . (1/20 - 1/81)
A = 5/2 . 61/1620
A = 61/648
B = 1/1.2.3 + 1/2.3.4 + ... + 1/18.19.29
2B = 2/1.2.3 + 2/2.3.4 + ... + 2/18.19.20
\(\Rightarrow\)B = 1/1.2 + 1/2.3 + ... + 1/19.20
\(\Rightarrow\)B = 1/1.2 - 1/19.20
B = 1/2 - 1/380
B = 189/380
\(M=\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{100.101.102}\right)\)
\(M=\frac{1}{2}.\left(1-\frac{1}{102}\right)\)
\(M=\frac{101}{204}< 1\left(đpcm\right)\)
Ta có: M=11.2.3 +12.3.4 +13.4.5 +...+1100.101.102
M=2.(11.2.3 +12.3.4 +13.4.5 +...+1100.101.102 ).12
M=(21.2.3 +22.3.4 +23.4.5 +...+2100.101.102 ).12
M=(11.2 -12.3 +12.3 -13.4 +13.4 -14.5 +...+1100.101 −1101.102 ).12
M=( 11.2 −1101.102 ).12
Mà 11.2 −1101.102 <1
Và 12 <1
=> (11.2 −1101.102 ) .12 <1
=> M <1
nhớ 9 k đó
\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{20.21.22}=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{20.21.22}\right)\)
\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+....+\frac{1}{20.21}-\frac{1}{21.22}\right)\)
\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{462}\right)=\frac{1}{2}.\frac{115}{231}=\frac{115}{462}\)
