Đặt Biểu thức trên là A

\(A=\frac{1}{2.5.9}+\frac{1}{2.9.13}+\frac{1}{2.13.17}+...+\frac{1}{2.397.401}+\frac{1}{2.401.405}\)

\(A=\frac{1}{2}\left(\frac{1}{5.9}+\frac{1}{9.13}+\frac{1}{13.17}+...+\frac{1}{397.401}+\frac{1}{401.405}\right)\)

\(4A=\frac{1}{2}\left(\frac{4}{5.9}+\frac{4}{9.13}+\frac{4}{13.17}+...+\frac{4}{397.401}+\frac{4}{401.405}\right)\)

\(4A=\frac{1}{2}\left(\frac{9-5}{5.9}+\frac{13-9}{9.13}+\frac{17-13}{13.17}+...+\frac{401-397}{397.401}+\frac{405-401}{401.405}\right)\)

\(4A=\frac{1}{2}\left(\frac{1}{5}-\frac{1}{405}\right)=\frac{1}{2}.\frac{80}{405}=\frac{40}{405}\Rightarrow A=\frac{40}{4.405}=\frac{2}{81}\)

a)Ta có: 

A= 1/2.9 + 1/9.7 +...+1/252.509

  = 2/5.(5/4.9 + 5/9.14 + 5/14.19 +...+ 1/504.509)

  = 2/5.(1/4 - 1/9 + 1/9 - 1/14 + 1/14 - 1/19 +...+ 1/504 - 1/509)

  = 2/5.(1/4 - 1/509)

  = 101/1018

Vậy A = 101/1018

b)Ta có:

B= 1/10.9 +1/18.13 + 1/26.17 +...+ 1/802.405)

  = 1/4.(8/10.18 + 8/18.26 + 8/26.34 +...+ 8/802.810)

  = 1/4.(1/10 - 1/18 + 1/18 - 1/26 + 1/26 - 1/34 +...+ 1/802 - 1/810)

  = 1/4.(1/10 - 1/810)

  = 2/81

Vậy B= 2/81

Tk mình nha!!!

a,A=101/1018 b,B=2/81

\(\frac{n}{n}\)

\(A=\frac{1}{2.9}+\frac{1}{9.7}+...+\frac{1}{252.509}\)

\(A=\frac{2}{4.9}+\frac{2}{9.14}+...+\frac{2}{504.509}\)

\(A=\frac{2}{5}.\left(\frac{5}{4.9}+\frac{5}{9.14}+...+\frac{5}{504.509}\right)\)

\(A=\frac{2}{5}.\left(\frac{9-4}{4.9}+\frac{14.9}{9.14}+...+\frac{509-504}{504.509}\right)\)

\(A=\frac{2}{5}.\left(\frac{1}{4}-\frac{1}{9}+\frac{1}{9}-\frac{1}{14}+...+\frac{1}{504}-\frac{1}{509}\right)\)

\(A=\frac{2}{5}.\left(\frac{1}{4}-\frac{1}{509}\right)\)

\(A=\frac{2}{5}.\frac{505}{2036}\)

\(A=\frac{101}{1018}\)

\(B=\frac{1}{10.9}+\frac{1}{18.13}+\frac{1}{26.17}+...+\frac{1}{802.405}\)

\(\frac{1}{2}B=\frac{1}{10.9.2}+\frac{1}{18.13}+\frac{1}{26.17}+...+\frac{1}{802.405.2}\)

\(\frac{1}{2}B=\frac{1}{10.18}+\frac{1}{18.26}+\frac{1}{26.34}+...+\frac{1}{802.810}\)

\(4B=\frac{8}{10.18}+\frac{8}{18.26}+\frac{8}{26.34}+...+\frac{8}{802.810}\)

\(4B=\frac{18-10}{10.18}+\frac{26-18}{28.26}+\frac{34-26}{26.34}+...+\frac{810-802}{802.810}\)

\(4B=\frac{1}{10}-\frac{1}{18}+\frac{1}{18}-\frac{1}{26}+\frac{1}{26}-\frac{1}{34}+...+\frac{1}{802}-\frac{1}{810}\)

\(4B=\frac{1}{10}-\frac{1}{810}\)

\(4B=\frac{8}{81}\)

\(B=\frac{2}{81}\)

\(C=\frac{2}{4.7}-\frac{3}{5.9}+\frac{2}{7.10}-\frac{3}{9.13}+...+\frac{2}{301.304}-\frac{3}{401.405}\)

\(C=\left(\frac{2}{4.7}+\frac{2}{7.10}+...+\frac{2}{301.304}\right)-\left(\frac{3}{5.9}+\frac{3}{9.13}+...+\frac{3}{401.405}\right)\)

\(C=\frac{2}{3}.\left(\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{301}-\frac{1}{304}\right)-\frac{3}{4}.\left(\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+...+\frac{1}{401}-\frac{1}{405}\right)\)

\(C=\frac{2}{3}.\left(\frac{1}{4}-\frac{1}{304}\right)-\frac{3}{4}.\left(\frac{1}{5}-\frac{1}{405}\right)\)

\(C=\frac{2}{3}.\frac{75}{304}-\frac{3}{4}.\frac{16}{81}\)

\(C=\frac{25}{152}-\frac{4}{27}\)

\(C=\frac{67}{4104}\)

7A=1/2-1/509=507/1018

C/2 = 1/10.18 + 1/18.26 + 1/26.34 + ....... + 1/802.810

4C = 8C/2 = 8/10.18 + 8/18.26 + 8/26.34 + ...... + 8/802.810

                 = 1/10 - 1/18 + 1/18 - 1/26 + 1/26 - 1/34 + ...... + 1/802 - 1/810

                 = 1/10 - 1/810 = 8/81

=> C = 8/81 : 4 = 2/81

Tk mk nha