\(B=1.2+2.3+3.4+...+18.19\)

\(3B=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+18.19.\left(20-17\right)\)

\(3B=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+18.19.20-17.18.19\)

\(3B=18.19.20\Rightarrow B=\frac{18.19.20}{3}=2280\)

Ta có :   B = 1 x 2 + 2 x 3 + 3 x 4 +.... + 18 x 19

=> 3B = 1 x 2 x 3 + 2 x 3 x 3 + 3 x 4 x 3 + ... + 18 x 19 x 3

     3B = 1 x 2 x 3 + 2 x 3 x (4 - 1) + 3 x 4 x (5 - 2) + .... + 18 x 19 x (20 - 17)

     3B = 1 x 2 x 3 + 2 x 3 x 4 - 1 x 2 x3 + 3 x 4 x5 - 2 x 3 x 4 + ... + 18 x 19 x 20 - 17 x 18 x 19

     3B = 18 x 19 x 20

       B = 18 x 19 x 20 : 3 

          = 2280

Vậy B = 2280 

trả lời:

\(B=1.2+2.3+3.4+...+18.19\)

\(3B=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+18.19.\left(20-17\right)\)

\(3B=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+18.19.20-17.18.19\)

\(3B=18.19.20\Rightarrow B=\frac{18.19.20}{3}=2280\)

học tốt

\(\dfrac{2}{1\cdot2}+\dfrac{2}{2\cdot3}+\dfrac{2}{3\cdot4}+...+\dfrac{2}{19\cdot20}\)

\(=2\cdot\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{19}-\dfrac{1}{20}\right)\)

\(=2\cdot\left(1-\dfrac{1}{20}\right)\)

\(=2\cdot\dfrac{19}{20}\)

\(=\dfrac{19}{10}\)

\(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{19\cdot20}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{19}-\frac{1}{20}\)

\(=\frac{1}{2}-\frac{1}{20}\)

\(=\frac{9}{20}\)

\(\frac{1}{2x3}\)+ \(\frac{1}{3x4}\)+ \(\frac{1}{4x5}\)+ ... + \(\frac{1}{18x19}\)+ \(\frac{1}{19x20}\)

= \(\frac{1}{2}\)- \(\frac{1}{3}\)+ \(\frac{1}{3}\)- \(\frac{1}{4}\)+ \(\frac{1}{4}\)- \(\frac{1}{5}\)+ ... + \(\frac{1}{18}\)- \(\frac{1}{19}\)+ \(\frac{1}{19}\)- \(\frac{1}{20}\)

= \(\frac{1}{2}\)- \(\frac{1}{20}\)

= \(\frac{18}{40}\)= \(\frac{9}{20}\)

=1/2-1/3+1/3-1/4+...+1/18-1/19+1/19-1/20         K MIK NHA MOI NGUOI 

=1/2-1/20

=10/20-1/20

=9/20

\(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+....+\frac{2}{19.20}\)

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

\(=2.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{19}-\frac{1}{20}\right)\)

\(=2.\left(1-\frac{1}{20}\right)\)

\(=2.\frac{19}{20}=\frac{19}{10}\)

Ta có: \(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{18.19}+\frac{2}{19.20}=2.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{18.19}+\frac{1}{19.20}\right)\)

      \(=2.\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\right)\)

       \(=2.\left(1-\frac{1}{20}\right)=\frac{2.19}{20}=\frac{19}{10}\)

\(\frac{2}{1\times2}+\frac{2}{2\times3}+......+\frac{2}{19\times20}\)

\(=2\left(\frac{1}{1\times2}+\frac{1}{2\times3}+.......+\frac{1}{19\times20}\right)\)

\(=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+........+\frac{1}{19}-\frac{1}{20}\right)\)

\(=2\left(1-\frac{1}{20}\right)=2.\frac{19}{20}=\frac{19}{10}\)

\(\frac{2}{1\times2}+\frac{2}{2\times3}+....+\frac{2}{19\times20}\)
\(=2\times\left(\frac{1}{1\times2}+\frac{1}{2\times3}+....+\frac{1}{19\times20}\right)\)
\(=2\times\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....+\frac{1}{19}-\frac{1}{20}\right)\)
\(=2\times\left(1-\frac{1}{20}\right) =2\times\frac{19}{20} =\frac{19}{10}\)
chúc bạn học tôt nha ^^

\(=\dfrac{1}{3}\left(1\times2\times3+2\times3\times3+...+19\times20\times3\right)\\ =\dfrac{1}{3}\left[1\times2\times\left(3-0\right)+2\times3\times\left(4-1\right)+...+19\times20\times\left(21-18\right)\right]\\ =\dfrac{1}{3}\left(1\times2\times3-1\times2\times3+2\times3\times4-...-18\times19\times20+19\times20\times21\right)\\ =\dfrac{1}{3}\times19\times20\times21=2660\)

Ta dễ dàng nhận thấy: \(\frac{1}{2\times3}=\frac{3-2}{2\times3}=\frac{3}{2\times3}-\frac{2}{2\times3}=\frac{1}{2}-\frac{1}{3}\).

Vậy, ta có thể tính dãy này như sau:

\(\frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}+...+\frac{1}{19\times20}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{19}-\frac{1}{20}\)

Ta gạch đi những phân số giống nhau và bằng nhau, Ta còn \(\frac{1}{2}\)và \(\frac{1}{20}\). Vậy từ đó ta có:

\(=\frac{1}{2}-\frac{1}{20}=\frac{9}{20}\)

\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{18.19}+\)\(\frac{1}{19.20}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\)

\(=\frac{1}{2}-\frac{1}{20}=\frac{10-1}{20}=\frac{9}{20}\)

\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{18.19}+\frac{1}{19.20}\)

\(=\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+...+\left(\frac{1}{18}-\frac{1}{19}\right)+\left(\frac{1}{19}-\frac{1}{20}\right)\)vì có \(\frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\)

\(=\frac{1}{2}-\frac{1}{20}=\frac{9}{20}\)

Đặt \(A=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+....+\frac{2}{18.19}+\frac{2}{19.20}\)

\(A=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\right)\)

\(A=2\left(1-\frac{1}{20}\right)\)

\(A=2.\frac{19}{20}=\frac{19}{10}\)

\(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{18.19}+\frac{2}{19.20}\)

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

\(=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\right)\)

\(=2\left(1-\frac{1}{20}\right)\)

\(=2.\frac{19}{20}\)

\(=\frac{19}{10}\)

Ta có : \(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+......+\frac{2}{19.20}\)

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

\(=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+......+\frac{1}{19}-\frac{1}{20}\right)\)

\(=2\left(1-\frac{1}{20}\right)=2.\frac{19}{20}=\frac{19}{10}\)