=1/2 + (1/2*3+1/3*4) + (1/4*5+1/5*6) + (1/6*7+1/7*8) + (1/8*9+1/9*10)
=1/2 + 1/2*3.(1+1/2) + 1/2*5.(1/2+1/3) + 1/2*7.(1/3+1/4) + 1/2*9.(1/4+1/5)
=1/2 + 1/2*3.(3/2) + 1/2*5.(5/6) + 1/2*7.(7/12) + 1/2*9.(9/20)
=1/2 + 1/4 + 1/12 + 1/24 + 1/40
=9/10

6 = 2 x 2 + 2 = 2(2+1)
12 = 3 x 3 + 3 = 3(3+1)
20 = 4 x 4 + 4 = 4(4+1)
...
90 = 9 x 9 + 9 = 9(9+1)

Ta có đồng nhất thức sau 
1/[n(n+1)] = 1/n - 1/(n+1)

Vậy
     1/6 = 1/2 - 1/3
     1/12 = 1/3 - 1/4
     1/20 = 1/4 - 1/5
....
     1/90 = 1/9 - 1/110

S =  1/2 -1/3+1/3-1/4+..............+1/9 -1/10

Tổng là 1/2 + 1/2 - 1/10 = 1 - 1/10 = 9/10
 

\(S=\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{90}\)

\(\Rightarrow S=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{9.10}\)

       \(S=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\)(áp dụng quy tắc dấu ngoặc )

       \(S=\frac{1}{2}-\left(\frac{1}{3}-\frac{1}{3}\right)-\left(\frac{1}{4}-\frac{1}{4}\right)-...-\left(\frac{1}{9}-\frac{1}{9}\right)-\frac{1}{10}\)

      \(S=\frac{1}{2}-\frac{1}{10}\) 

      \(S=\frac{2}{5}\)

S= \(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}\)

S= 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 +  1/5 -1/6 +1/6 - 1/7 + 1/7 - 1/8

S= 1/2 - 1/ 8 

S= 3/8

S= 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/6.7 + 1/7.8

  = 1/2 - 1/3 + 1/3 - ...+ 1/7 - 1/8

  = 1/2 - 1/8

  = 3/8

S=(1/2.3) +(1/3.4) + (1/4.5) +...+1/7.8

S=1/2-1/3+1/3-1/4 +1/4-1/5+...+1/7-1/8

S=1/2-1/8

S=3/8

S=2/3.7+2/7.1+2/11.15+...+2/95.99

S = \(\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{7}\right)+\frac{1}{2}.\left(\frac{1}{7}-\frac{1}{11}\right)+\frac{1}{2}.\left(\frac{1}{11}-\frac{1}{15}\right)+...+\frac{1}{2}.\left(\frac{1}{95}-\frac{1}{99}\right)\)

S = \(\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{95}-\frac{1}{99}\right)\)

S = \(\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{99}\right)\)

S = \(\frac{1}{2}.\frac{32}{99}\)

S = \(\frac{16}{99}\)

M=1/2+1/6+1/12+1/20+..+1/110

M = \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{10.11}\)

M = \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{10}-\frac{1}{11}\)

M = \(1-\frac{1}{11}\)

M = \(\frac{10}{11}\)

\(S=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{45.46}\)

\(\Rightarrow S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{45.46}\)

\(\Rightarrow S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{45}-\frac{1}{46}\)

\(\Rightarrow S=1-\frac{1}{46}\)

\(\Rightarrow S=\frac{45}{46}\)

Bài làm

\(S=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{45.46}\)

\(S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{45.46}\)

\(S=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{45}-\frac{1}{46}\)

\(S=\frac{1}{1}-\frac{1}{46}\)

\(S=\frac{46}{46}-\frac{1}{46}\)

\(S=\frac{45}{46}\)

Vậy \(S=\frac{45}{46}\)

# Học tốt #

\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{650}\)  = \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{25.26}\)= \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{25}-\frac{1}{26}\)= \(1-\frac{1}{26}=\frac{25}{26}\)

~Học tốt~

\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{650}\)

\(=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{25.26}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{25}-\frac{1}{26}\)

\(=1-\frac{1}{26}=\frac{25}{26}\)

1/2+1/6+1/12+....+1/650

=1/1.2+1/2.3+....+1/25.26

=1-1/2+1/2+.....+1/25-1/26

=1-1/26

=25/26

Giải hộ mik nha mấy bạn

\(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{90}\)

\(\frac{1}{6}+\frac{1}{12}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{90}\)

\(=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{9.10}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-...+\frac{1}{9}-\frac{1}{10}\)

\(=1-\frac{1}{10}\)

\(=\frac{9}{10}\)

 S= 1/2 + 1/6 + 1/12 + ...+ 1/90

S=1/1.2 + 1/2.3 +1/3.4+...+1/9.10

S=1-1/2 + 1/2-1/3 + 1/3-1/4+...+1/9-1/10

S=(1-1/10)+(1/2-1/2)+(1/3-1/3)+...+(1/9-1/9)

S=9/10+0+0+...+0

S=9/10Chúc bn học tốt =))