Question

Tính tổng s= 1/6+ 1/12+ 1/20+ ........+ 1/90

Answer 1

=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

Answer 2

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
 

Answer 3

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

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

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

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

      \(S=\frac{1}{2}-\frac{1}{10}\) 

      \(S=\frac{2}{5}\)