Hình như bạn chép sai đề, mình sửa nhé :

\(S=\dfrac{5^2}{1.6}+\dfrac{5^2}{6.11}+\dfrac{5^2}{11.16}+...+\dfrac{5^2}{26.31}\\ =>\dfrac{S}{5}=\dfrac{5}{1.6}+\dfrac{5}{6.11}+\dfrac{5}{11.16}+...+\dfrac{5}{26.31}\\ =>\dfrac{S}{5}=\dfrac{6-1}{1.6}+\dfrac{11-6}{6.11}+\dfrac{16-11}{11.16}+...+\dfrac{31-26}{26.31}\\ =>\dfrac{S}{5}=1-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{16}+...+\dfrac{1}{26}-\dfrac{1}{31}=1-\dfrac{1}{31}=\dfrac{30}{31}\\ =>S=\dfrac{30}{31}.5=\dfrac{150}{31}\)

\(=5\left(\dfrac{5}{1\cdot6}+\dfrac{5}{6\cdot11}+...+\dfrac{5}{101\cdot106}\right)\\ =5\left(1-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+...+\dfrac{1}{101}-\dfrac{1}{106}\right)\\ =5\left(1-\dfrac{1}{106}\right)=5\cdot\dfrac{105}{106}=\dfrac{525}{106}\)

Gọi 3 phần dc chia thành là x,y,z

=> 2x =3y =4z ; x+y+z =52

\(\frac{x}{\frac{1}{2}}=\frac{y}{\frac{1}{3}}=\frac{z}{\frac{1}{4}}=\frac{x+y+z}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}}=\frac{52}{\frac{13}{12}}=48\)

=>.x = 48.1/2 =24

=>y =48.1/3 =16

=>z = 48.1/4 =12

\(B=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+\frac{1}{18\cdot19\cdot20}\)

\(B=\frac{1}{2}\left(\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+\frac{2}{18\cdot19\cdot20}\right)\)

\(B=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{18\cdot19}-\frac{1}{19\cdot20}\right)\)

\(B=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{19\cdot20}\right)\)

\(B=\frac{1}{2}\cdot\frac{189}{380}=\frac{189}{760}\)

\(C=\frac{52}{1\cdot6}+\frac{52}{6\cdot11}+\frac{52}{11\cdot16}+...+\frac{52}{31\cdot36}\)

\(C=\frac{52}{5}\left(\frac{5}{1\cdot6}+\frac{5}{6\cdot11}+\frac{5}{11\cdot16}+...+\frac{6}{31\cdot36}\right)\)

\(C=\frac{52}{5}\left(1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+...+\frac{1}{31}-\frac{1}{36}\right)\)

\(C=\frac{52}{5}\cdot\left(1-\frac{1}{36}\right)\)

\(C=\frac{91}{9}\)

\(Q=\frac{5^2}{1.6}+\frac{5^2}{6.11}+\frac{5^2}{11.16}+...+\frac{5^2}{26.31}\)

\(=5\left(\frac{5}{1.6}+\frac{5}{6.11}+\frac{5}{11.16}+...+\frac{5}{26.31}\right)\)

\(=5\left(1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+...+\frac{1}{26}-\frac{1}{31}\right)\)

\(=5\left(1-\frac{1}{31}\right)\)

\(=5\cdot\frac{30}{31}=\frac{150}{31}\)

Bạn nhân 2 lên rồi áp dụng \(\frac{5}{a\times\left(a+5\right)}=\frac{1}{a}-\frac{1}{a+5}\) thì sẽ còn lại là    2Q=1-1/31 =30/31 nên Q=30/62

TS hơn MS : 91-52=39 ( bn tự vẽ sơ đồ nha)

TS số của phân số mới là : 39 : 1 *2  =78

Vay phai bot ca TS va MS di : 78-52=26

              ĐÁP SỐ : 26

E=\(\frac{10}{1\cdot6}\) +\(\frac{10}{6\cdot11}\) +\(\frac{10}{11\cdot16}\) +\(\frac{10}{16\cdot21}\) +\(\frac{10}{21\cdot26}\) +\(\frac{10}{26\cdot31}\)                                                                                      =  5*(1-\(\frac{1}{31}\) )                                                                                                                                                                                 =5*\(\frac{30}{31}\)        =\(\frac{150}{31}\) 

Có: \(A=\frac{5^2}{1.6}+\frac{5^2}{6.11}+...+\frac{5^2}{26.31}\)

\(=5.\left(\frac{6-1}{1.6}+\frac{11-6}{6.11}+...+\frac{31-26}{26.31}\right)\)

\(=5.\left(\frac{6}{1.6}-\frac{1}{1.6}+\frac{11}{6.11}-\frac{6}{6.11}+...+\frac{31}{26.31}-\frac{26}{26.31}\right)\)

\(=5.\left(1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+...+\frac{1}{26}-\frac{1}{31}\right)\)

\(=5.\left(1-\frac{1}{31}\right)\)

\(=5.\frac{30}{31}=\frac{150}{31}>\frac{31}{31}=1\)

\(\Rightarrow A>1\)

Ta có: A=\(\frac{5^2}{1.6}\)+\(\frac{5^2}{6.11}\)+...+\(\frac{5^2}{26.31}\)

=5.(\(\frac{5}{1.6}\)+\(\frac{5}{6.11}\)+...+\(\frac{5}{26.31}\))

=5.(1-\(\frac{1}{6}\)+\(\frac{1}{6}\)-\(\frac{1}{11}\)+\(\frac{1}{11}\)+...+\(\frac{1}{26}\)-\(\frac{1}{30}\))

=5.(1-\(\frac{1}{30}\))

=5.\(\frac{29}{30}\)

=\(\frac{29}{6}\)>1

Hay A>1

=> đpcm