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 = \(\frac{19}{20}\)

a=(1/1-1/2) + (1/2 +1/3) +(1/3+1/4)+........+(1/2016+1/2017)

a=1/1-1/2+1/2-1/3+1/3-1/4+ ........+2016-2017

a=1/1+1/2017 =2017-1/2017

a=2016/2017

chac chan 100% do ban .tk mk nha

99/100 k nha

999/100

99/100 nha! >_<

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tham khảo nhé, vội quá ko trình bày cho bạn được

Ta có: \(\frac{-3}{1.2.3}+\frac{-3}{2.3.4}+\frac{-3}{3.4.5}+...+\frac{-3}{18.19.20}\)

          \(=\frac{-3}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{18.19.20}\right)\)

          \(=\frac{-3}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

            \(=\frac{-3}{2}\left(\frac{1}{2}-\frac{1}{19.20}\right)=\frac{-3}{2}.\frac{189}{380}=\frac{-567}{760}\)

Ta có: \(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+.........+\frac{2}{18.19}+\frac{2}{19.20}\)
          = \(\frac{2}{1}-\frac{2}{2}+\frac{2}{2}-\frac{2}{3}+\frac{2}{3}-\frac{2}{4}+.......+\frac{2}{18}-\frac{2}{19}+\frac{2}{19}-\frac{2}{20}\)
          =\(\frac{2}{1}-\frac{2}{20}=\frac{40}{20}-\frac{2}{20}=\frac{38}{20}=\frac{19}{10}\).

B= 1/ 1.2.3 + 1/ 2.3 4 + 1/ 3.4.5 + .... + 1/ 18.19.20

Ta có:

1/ 1.2 - 1/ 2.3 = 2/ 1.2.3

1/ 2.3 - 1/3.4 = 2/ 2.3.4

Từ đó Ta có: B = 1/2 . ( 2/ 1.2.3 + 2/ 2,3.4 + ... + 2/ 18. 19. 20 )

= 1/2 .( 1/ 1.2 – 1/ 2.3 + 1/ 2.3 - .....- 1/19.20)

= 1/2. ( 1/ 1.2 – 1/ 19.20 ) = 1/ 2 . 189/380 = 189/760

\(B=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+....+\frac{1}{18\cdot19\cdot20}\)

\(B=\frac{3-1}{1\cdot2\cdot3}+\frac{4-2}{2\cdot3\cdot4}+...+\frac{20-18}{18\cdot19\cdot20}\)

\(2B=\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+...+\frac{2}{18\cdot19\cdot20}\)

\(2B=\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+...+\frac{1}{18\cdot19}-\frac{1}{19\cdot20}\)

\(2B=\frac{1}{1\cdot2}-\frac{1}{19\cdot20}\)

\(\Rightarrow B=\left(\frac{1}{1\cdot2}-\frac{1}{19\cdot20}\right)\div2=\frac{189}{380}\div2=\frac{189}{760}\)

\(E=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)

\(E=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

\(E=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{19.20}\right)=\frac{1}{4}-\frac{1}{2.19.20}< \frac{1}{4}\left(đpcm\right)\)

Ta có : 

\(E=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\)

\(E=\frac{1}{2}-\frac{1}{380}=\frac{189}{380}< \frac{95}{380}=\frac{1}{4}\)

Vậy \(E< \frac{1}{4}\)

Chúc bạn học tốt ~

Àk nhầm rùi : 

Ta có : 

\(E=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)

\(\Leftrightarrow\)\(E=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\right)\)

\(\Leftrightarrow\)\(E=\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

\(\Leftrightarrow\)\(E=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{19.20}\right)=\frac{1}{4}-\frac{1}{2.19.20}< \frac{1}{4}\)

Vậy \(E< \frac{1}{4}\)

Học tốt :)

a)\(\frac{13}{15}+\frac{13}{35}+\frac{13}{63}+\frac{13}{99}\)

\(=\frac{13}{3.5}+\frac{13}{5.7}+\frac{13}{7.9}+\frac{13}{9.11}\)

\(=\frac{13}{2}\left(\frac{1}{3}-\frac{1}{5}+.....+\frac{1}{9}-\frac{1}{11}\right)\)

\(=\frac{13}{2}\left(\frac{1}{3}-\frac{1}{11}\right)\)

\(=\frac{13}{2}\cdot\frac{8}{33}\)

\(=\frac{52}{33}\)

a) Đặt A= 13/15 + 13/35 + 13/63 + 13/99

A = 13/2 ( 2/15 + 2/35 + 2/63 + 2/99)

A= 13/2 ( 2/ 3.5 + 2/5.7 + 2/7.9 + 2/9.11)

A= 13/2 ( 1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/11)

A= 13/2 ( 1/3 - 1/11) 

A= 13/2 . 8/33

A= 52/33  

\(b,\)\(\left(\frac{15}{1.2.3}+\frac{15}{2.3.4}+\frac{15}{3.4.5}+...+\frac{15}{18.19.20}\right).x=1\)

\(\left[\frac{15}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{15}{18.19.20}\right)\right].x=1\)

\(\left[\frac{15}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+\frac{1}{4.5}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\right].x=1\)

\(\left[\frac{15}{2}.\left(\frac{1}{1.2}-\frac{1}{19.20}\right)\right].x=1\)

\(\left[\frac{15}{2}.\frac{189}{380}\right].x=1\)

\(\frac{567}{152}.x=1\)

\(x=1-\frac{567}{152}\)

\(\Rightarrow x=-\frac{415}{152}\)