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   \(\left(1-\frac{1}{2}\right).\left(1-\frac{1}{3}\right).\left(1-\frac{1}{4}\right)......\left(1-\frac{1}{2017}\right)\)

\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}........\frac{2016}{2017}\)

\(=\frac{1}{2017}\)

=> 3B = 3.( 1/2.5 + 1/5.8 + 1/8.11 + ........... + 1/122.125)

           = 3/2.5 + 3/5.8 + 3/ 8.11 + ......+ 3/122.125

Ta có: 3/ 2.5 = 1/2 - 1/5 

          3/5.8  = 1/5 -1/8

          3/ 8.11 = 1/8 -1/11

          ..........................

         3/122 . 125 = 3/122 - 3/125

=> 3B=  1/2 - 15/5 + 1/5 -1/8 +1/8 - 1/11 +........+1/122 - 1/125

         =  1/2 - 1/125 = 125/250 - 2/250= 123/250

=> B= 3B : 3 = 123/250 :3 = 123/250 . 1/3 = 41/250

=> 2C = 2.(1/9.11 + 1/11.13 +....+ 1/97 .99)

           = 2/9.11 + 2/11 .13 +.....+ 2/ 97.99

Ta có: 2/9.11 = 1/9 - 1/11

          2/11.13 = 2/11 -2/ 13

         ...............................

         2/97.99 = 1/97 - 1/99

=> 2B = 1/9 - 1/11 + 1/11 - 1/13 + ....+ 1/97 - 1/99

           = 1/9 -1/99 = 11/99 - 1/99 =10/99

=> B= 2B : B = 10/99 :2 =10/99 . 1/2 = 5/99

Vậy B = 5/99

 \(B=\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+...+\frac{1}{122.125}\)

\(3B=\frac{3}{2.5}+\frac{3}{5.8}+\frac{3}{8.11}+...+\frac{3}{122.125}\)

Nhận xét:

\(\frac{3}{2.5}=\frac{1}{2}-\frac{1}{5}=\frac{3}{10}\)

\(\frac{3}{5.8}=\frac{1}{5}-\frac{1}{8}=\frac{3}{40}\)

\(\frac{3}{8.11}=\frac{1}{8}-\frac{1}{11}=\frac{3}{88}\)

.............

Từ nhận xét trên ta có:

\(3B=\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+...+\frac{1}{122}-\frac{1}{125}\)

\(3B=\frac{1}{2}-\frac{1}{125}=\frac{123}{250}\)

\(B=\frac{123}{250}:3=\frac{41}{250}\)

\(C=\frac{1}{9.11}+\frac{1}{11.13}+...+\frac{1}{97.99}\)

\(2C=\frac{2}{9.11}+\frac{2}{11.13}+...+\frac{2}{97.99}\)

Nhận xét:

\(\frac{2}{9.11}=\frac{1}{9}-\frac{1}{11}=\frac{2}{99}\)

\(\frac{2}{11.13}=\frac{1}{11}-\frac{1}{13}=\frac{1}{143}\)

..................

Từ nhận xét trên ta có:

\(2C=\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13}+...+\frac{1}{97}-\frac{1}{99}\)

\(2C=\frac{1}{9}-\frac{1}{99}=\frac{10}{99}\)

\(C=\frac{10}{99}:2=\frac{5}{99}\)

( 1-1/2) . (1-1/3).(1-1/4).......(1-1/2016) . (1-1/2017)

=1/2.2/3.3.4x...x2015/2016.2016/2017

=1.2.3.4. ... .2015.2016/2.3.4.5. ... .2016.2017

(giống nhau bạn gạch đi )

=1/2017

 ( 1-1/2) . (1-1/3).(1-1/4).......(1-1/2016) . (1-1/2017)

=1/2 . 2/3 . 3/4 ......... 2015/2016  .  2016/2017

=1.2 .3......2015.2016/2.3.4.......2016.2017

=1/2017

 =1/2 * 2/3 * 3/4 * ... * 2016/2017

= 1/2017

sai roi 0 co 1/30

\(\frac{1}{12}+\frac{1}{20}+\frac{1}{42}+.....+\frac{1}{132}\)

\(=\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+....+\frac{1}{11.12}\)

\(=\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+......+\frac{1}{11}-\frac{1}{12}\)

\(=\frac{1}{3}-\frac{1}{12}\)

\(=\frac{1}{4}\)

\(S=\frac{1}{12}+\frac{1}{20}+\frac{1}{42}+....+\frac{1}{132}\)

\(=\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+....+\frac{1}{11.12}\)

\(=\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{11}-\frac{1}{12}\)

\(=\frac{1}{3}-\frac{1}{12}=\frac{4-1}{12}=\frac{3}{12}=\frac{1}{4}\)

Khó quá

1/2 + 1/4 + 1/8 +1/16 + 1/32 + 1/64 + 1/128 

= 2 . ( 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 )

= 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 - 1/128 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128  ( Rồi giản ước )

= 1

\(B=\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{132}\)

\(B=\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\frac{1}{6\cdot7}+...+\frac{1}{11\cdot12}\)

\(B=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{11}-\frac{1}{12}\)

\(B=\frac{1}{4}-\frac{1}{12}\)

\(B=\frac{1}{6}\)

Tính nhanh : 

B = 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 + 1/110 + 1/132 

B = 1/4.5 + 1/5.6 + 1/6.7 + 1/7.8 + 1/8.9 + 1/9.10 + 1/10.11 + 1/11.12

B = 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7 + 1/7 - 1/8 + ...... + 1/11 - 1/12

B = 1/4 - 1/12

B = 3/12  - 1/12

B = 2/12

B = 1/6