đặt A=(1-1/3)........

Ta có A=\(\frac{2}{3}\cdot\frac{5}{6}\cdot\frac{9}{10}\cdot...\cdot\frac{209}{210}=\frac{4}{6}\cdot\frac{10}{12}\cdot\frac{18}{20}\cdot...\cdot\frac{418}{420}=\frac{1\cdot4}{2\cdot3}\cdot\frac{2\cdot5}{3\cdot4}\cdot\frac{3\cdot6}{4\cdot5}\cdot...\cdot\frac{19\cdot22}{20\cdot21}\)

=\(\frac{1\cdot4\cdot2\cdot5\cdot3\cdot6\cdot...\cdot19\cdot22}{2\cdot3\cdot3\cdot4\cdot4\cdot5\cdot...\cdot20\cdot21}=\frac{\left(1\cdot2\cdot3\cdot...\cdot19\right)\cdot\left(4\cdot5\cdot6\cdot...\cdot22\right)}{\left(2\cdot3\cdot4\cdot...\cdot20\right)\cdot\left(3\cdot4\cdot5\cdot...\cdot21\right)}\)

=\(\frac{1\cdot22}{20\cdot3}=\frac{11}{30}\)

Đặt \(A=\left(1-\frac{1}{3}\right).\left(1-\frac{1}{6}\right).\left(1-\frac{1}{10}\right).\left(1-\frac{1}{15}\right)...\left(1-\frac{1}{210}\right)\)

=>\(A=\frac{2}{3}.\frac{5}{6}.\frac{9}{10}.\frac{14}{15}...\frac{209}{210}\)

=>\(A=\frac{2.2}{3.2}.\frac{5.2}{6.2}.\frac{9.2}{10.2}.\frac{14.2}{15.2}...\frac{209.2}{210.2}\)

=>\(A=\frac{4}{6}.\frac{10}{12}.\frac{18}{20}.\frac{28}{30}...\frac{418}{420}\)

=>\(A=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}.\frac{4.7}{5.6}...\frac{19.22}{20.21}\)

=>\(A=\frac{\left(1.4\right).\left(2.5\right).\left(3.6\right).\left(4.7\right)...\left(19.22\right)}{\left(2.3\right).\left(3.4\right).\left(4.5\right).\left(5.6\right)...\left(20.21\right)}\)

=>\(A=\frac{\left(1.2.3.4...19\right).\left(4.5.6.7...22\right)}{\left(2.3.4.5...20\right).\left(3.4.5.6...21\right)}\)

=>\(A=\frac{1.22}{20.3}\)

=>\(A=\frac{22}{60}=\frac{11}{30}\)

Vậy \(\left(1-\frac{1}{3}\right).\left(1-\frac{1}{6}\right).\left(1-\frac{1}{10}\right).\left(1-\frac{1}{15}\right)...\left(1-\frac{1}{210}\right)=\frac{11}{30}\)

11/30 là đúng bạn ạ k hộ cái nha

C = \(\left(1-\frac{1}{3}\right)\left(1-\frac{1}{6}\right)\left(1-\frac{1}{10}\right)...\left(1-\frac{1}{210}\right)=\frac{2}{3}.\frac{5}{6}.\frac{9}{10}...\frac{209}{210}=\frac{4}{6}.\frac{10}{12}.\frac{18}{20}...\frac{418}{420}\)

= \(\frac{2.2}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}...\frac{19.22}{20.21}=\frac{2.2\left(2.3.4...19\right)\left(5.6...22\right)}{\left(2.3.4..20\right)\left(3.4.5..21\right)}=\frac{4.22}{19.3.4}=\frac{22}{57}\)

\(B=\left(1-\frac{1}{3}\right)\left(1-\frac{1}{6}\right)\left(1-\frac{1}{10}\right)\left(1-\frac{1}{15}\right)...\left(1-\frac{1}{210}\right)\)

\(B=\frac{2}{3}.\frac{5}{6}.\frac{9}{10}.\frac{14}{15}...\frac{209}{210}\)

\(B=\frac{1}{3}.\frac{1}{3}.\frac{9}{2}.\frac{14}{15}...\frac{209}{210}\)

\(B=\frac{1}{6}.\frac{9}{2}.\frac{14}{15}...\frac{209}{210}\)

\(B=\frac{1}{2}.\frac{1}{1}.\frac{7}{5}...\frac{209}{210}\)

\(B=\frac{7}{10}...\frac{209}{210}\)

\(B=\frac{62}{210}\)

Lời giải:

$\frac{A}{2}=\frac{1}{2}-\frac{1}{6}-\frac{1}{12}-\frac{1}{20}-...-\frac{1}{420}$

$\frac{A}{2}=\frac{1}{2}-\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{420}\right)$

Xét:

$\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{420}=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{20.21}$

$=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{20}-\frac{1}{21}$

$=\frac{1}{2}-\frac{1}{21}$

Do đó:

$\frac{A}{2}=\frac{1}{2}-(\frac{1}{2}-\frac{1}{21})=\frac{1}{21}$

$\Rightarrow A=\frac{2}{21}$

S1/2=1/2+1/6+1/12+...+1/600

S1/2=1/1*2+1/2*3+....+1/24*25

S1/2=1-1/25

S1/2=24/25

S=48/25

C=2/3.5/6.9/10...209/210

C=4/6.10/12.18/20...418/420 là do nhân với 2

C=1.4/2.3.2.5/3.4.3.6/4.5...19.22/20.21

C=1.2.3....19/2.3.4...20.4.5.6...22/3.4.5...21

C=1/20.22/3

C=11/30

Dễ ấy mà hiểu chưa