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

\(=1+\frac{1}{2}\times\frac{2.3}{2}+\frac{1}{3}\times\frac{3.4}{2}+...+\frac{1}{20}\times\frac{20.21}{2}\)

\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+...+\frac{21}{2}\)

\(=\frac{\left(2+21\right).20:2}{2}=\frac{230}{2}=115\)

Số cuối là

\(\frac{1}{10}.\left(1+2+3+...+10\right)\) hay \(\frac{1}{20}.\left(1+2+3+...+20\right)\) ??

\(=\dfrac{8\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)}{2}-\dfrac{3^{16}}{2}\)

\(=\dfrac{\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)-3^{16}}{2}\)

\(=\dfrac{\left(3^8-1\right)\left(3^8+1\right)-3^{16}}{2}\)

=-1/2

1/1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6

= 1/1-1/6

=6/6-1/6

= 5/6

=5/6 nhé.

chúc Phạm Quốc Cường học tốt nhé

1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 -1/6

= 1/1 - 1/6

= 5/6 

a,  \(2\dfrac{1}{3}+4\dfrac{1}{5}+4\dfrac{1}{3}\)

\(=\dfrac{7}{3}+\dfrac{21}{5}+\dfrac{13}{3}\)

\(=\dfrac{1}{3}\left(7+13\right)+\dfrac{21}{5}\)

\(=\dfrac{20}{3}+\dfrac{21}{5}=\dfrac{100+63}{15}=\dfrac{163}{15}\)

b, \(5\dfrac{3}{4}-4\dfrac{1}{2}.3\dfrac{7}{8}\)

\(=\dfrac{23}{4}-\dfrac{9}{2}.\dfrac{31}{8}\)

\(=\dfrac{23}{4}-\dfrac{279}{16}=\dfrac{92-279}{16}=-\dfrac{187}{16}\)

c, \(1\dfrac{1}{2}.3\dfrac{2}{3}.4\dfrac{3}{4}\)

\(=\dfrac{3}{2}.\dfrac{11}{3}.\dfrac{19}{4}=\dfrac{209}{8}\)

d, \(6\dfrac{4}{5}:2\dfrac{3}{4}:1\dfrac{1}{2}\)

\(=\dfrac{34}{5}:\dfrac{11}{4}:\dfrac{3}{2}\)

\(=\left(\dfrac{34}{5}.\dfrac{4}{11}\right).\dfrac{2}{3}=\dfrac{136}{55}.\dfrac{2}{3}=\dfrac{272}{165}\)

1.a 

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

= 1/2+1/2-1/4+1/4-1/8+1/8-1/16+1/16-1/32

= 1-1/32=31/32

1b

\(\frac{1}{2}.\frac{1}{2}+\frac{1}{2}.\frac{1}{3} +\frac{1}{3}+\frac{1}{3}+\frac{1}{4}+\frac{1}{4}.\frac{1}{5}+\frac{1}{5}.\frac{1}{6}\)

\(=\frac{1}{4}+\frac{1}{6}+\frac{2}{3}+\frac{1}{4}+\frac{1}{20}+\frac{1}{30}\)

\(=\frac{5}{20}+\frac{5}{30}+\frac{20}{30}+\frac{5}{20}+\frac{1}{20}+\frac{1}{30}\)

\(=\left(\frac{5}{20}+\frac{5}{20}+\frac{1}{20}\right)+\left(\frac{5}{30}+\frac{20}{30}+\frac{1}{30}\right)\)

\(=\frac{11}{20}+\frac{26}{30}\)

\(=\frac{11}{20}+\frac{13}{15}\)

\(=\frac{17}{12}\)

Đó là các cách bạn Võ Thạch Tín đã nêu ra nên mình ko còn câu trả lời nào hết

\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}\)

\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}\)

\(\frac{1}{2}-\frac{1}{6}=\frac{1}{3}\)

Tính nhanh  : 

1/2*3  + 1/3*4 + 1/4*5 + 1/5*6 

= 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 

= 1/2 - 1/6 

= 6/12 - 2/12 

= 4/12 

= 1/3

\(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}\)

= \(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}\)

= \(\frac{1}{2}-\frac{1}{6}\)

= \(\frac{3}{6}-\frac{1}{6}\)= \(\frac{2}{6}\)= \(\frac{1}{3}\)