. Tham khảo nha bạn: https://olm.vn/hoi-dap/question/894463.html

Đề sai. câu đầu phải là \(\dfrac{1}{18.21}\) mới đúng.
Nếu câu đầu là \(\dfrac{1}{18.21}\) thì mik có cách làm sau :
\(\dfrac{1}{18.21}+\dfrac{1}{21.24}+\dfrac{1}{24.27}+...+\dfrac{1}{123.126}\)

= \(\dfrac{1}{3}.\left(\dfrac{3}{18.21}+\dfrac{3}{21.24}+\dfrac{3}{24.27}+...+\dfrac{3}{123.126}\right)\)

= \(\dfrac{1}{3}.\left(\dfrac{1}{18}-\dfrac{1}{21}+\dfrac{1}{21}-\dfrac{1}{24}+\dfrac{1}{24}-\dfrac{1}{27}+...+\dfrac{1}{123}-\dfrac{1}{126}\right)\)

= \(\dfrac{1}{3}.\left(\dfrac{1}{18}-\dfrac{1}{126}\right)\)

= \(\dfrac{1}{3}.\dfrac{1}{21}\)

= \(\dfrac{1}{63}\)

\(D=\dfrac{3}{18\cdot21}+\dfrac{3}{21\cdot24}+\dfrac{3}{24\cdot27}+...+\dfrac{3}{123\cdot126}\\ =\dfrac{1}{18}-\dfrac{1}{21}+\dfrac{1}{21}-\dfrac{1}{24}+\dfrac{1}{24}-\dfrac{1}{27}+...+\dfrac{1}{123}-\dfrac{1}{126}\\ =\dfrac{1}{18}-\dfrac{1}{126}\\ =\dfrac{1}{21}\)

\(A=\frac{5}{18.21}+\frac{5}{21.24}+\frac{5}{24.27}+...+\frac{5}{123.126}\)

\(A=\frac{5}{3}\left(\frac{3}{18.21}+\frac{3}{21.24}+\frac{3}{24.27}+...+\frac{3}{123.126}\right)\)

\(A=\frac{5}{3}\left(\frac{1}{18}-\frac{1}{21}+\frac{1}{21}-\frac{1}{24}+\frac{1}{24}-\frac{1}{27}+...+\frac{1}{123}-\frac{1}{126}\right)\)

\(A=\frac{5}{3}\left(\frac{1}{18}-\frac{1}{126}\right)\)

\(A=\frac{5}{3}.\frac{1}{21}\)

\(A=\frac{5}{63}\)

3/5*A=3/(18*21)+3/(21*24)+3/(24*27)+...+3/(123*126)

=>3/5*A=1/18-1/21+1/21-1/24+1/24-1/27+...+1/123-1/126

=>3/5*A=1/18-1/126

=>3/5*A=1/21

=>A=5/63

A=\(\frac{5}{3}\left(\frac{3}{18.21}+\frac{3}{21.24}+..........+\frac{3}{123.126}\right)\)

A=\(\frac{5}{3}\left(\frac{1}{18}-\frac{1}{21}+\frac{1}{21}-\frac{1}{24}+............+\frac{1}{123}-\frac{1}{126}\right)\)

=\(\frac{5}{3}.\left(\frac{1}{18}-\frac{1}{126}\right)\)

=\(\frac{5}{3}.\frac{1}{21}\)

=\(\frac{5}{63}\)

1,

\(\dfrac{3}{2^2}\cdot\dfrac{8}{3^2}\cdot\dfrac{15}{4^2}...\dfrac{899}{30^2}\\ =\dfrac{1\cdot3}{2\cdot2}\cdot\dfrac{2\cdot4}{3\cdot3}\cdot\dfrac{3\cdot5}{4\cdot4}....\dfrac{29\cdot31}{30\cdot30}\\ =\left(\dfrac{1\cdot2\cdot3\cdot...\cdot29}{2\cdot3\cdot4\cdot....\cdot30}\right)\cdot\left(\dfrac{3\cdot4\cdot5\cdot....\cdot31}{2\cdot3\cdot4.....\cdot30}\right)\\ =\dfrac{1}{30}\cdot\dfrac{31}{2}\\ =\dfrac{31}{60}\)

2,

\(\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot4}+\dfrac{1}{3\cdot4\cdot5}+...+\dfrac{1}{37\cdot38\cdot39}\\ =\dfrac{1}{2}\left(\dfrac{2}{1\cdot2\cdot3}+\dfrac{2}{2\cdot3\cdot4}+\dfrac{2}{3\cdot4\cdot5}+...+\dfrac{2}{37\cdot38\cdot39}\right)\\ =\dfrac{1}{2}\left(\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+\dfrac{1}{3\cdot4}-\dfrac{1}{4\cdot5}+....+\dfrac{1}{37\cdot38}-\dfrac{1}{38\cdot39}\right)\\ =\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{38\cdot39}\right)\\ =\dfrac{1}{4}-\dfrac{1}{3964}\\ =\dfrac{185}{741}\)

3, Làm tương tự, áp dụng ; \(\dfrac{n}{x\left(x+n\right)}=\dfrac{1}{x}-\dfrac{1}{x+n}\)

a) \(\dfrac{5}{1\cdot6}+\dfrac{5}{6\cdot11}+...+\dfrac{5}{26\cdot31}\)

\(=1-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+...+\dfrac{1}{26}-\dfrac{1}{31}\)

\(=1-\dfrac{1}{31}\)

\(=\dfrac{30}{31}\)

b) \(\dfrac{4}{11\cdot16}+\dfrac{4}{16\cdot21}+...+\dfrac{4}{61\cdot66}\)

\(=\dfrac{4}{5}\cdot\left(\dfrac{5}{11\cdot16}+\dfrac{5}{16\cdot21}+...+\dfrac{5}{61\cdot66}\right)\)

\(=\dfrac{4}{5}\cdot\left(\dfrac{1}{11}-\dfrac{1}{16}+\dfrac{1}{16}-...+\dfrac{1}{61}-\dfrac{1}{66}\right)\)

\(=\dfrac{4}{5}\cdot\left(\dfrac{1}{11}-\dfrac{1}{66}\right)\)

\(=\dfrac{4}{5}\cdot\dfrac{5}{66}\)

\(=\dfrac{4}{66}\)

\(=\dfrac{2}{33}\)

a) A = 5²/(1.6) + 5²/(6.11) + ... + 5²/(26.31)

= 5.[5/(1.6) + 5/(6.11) + ...+ 5/(26.31)]

= 5.(1 - 1/6 + 1/6 - 1/11 + ... + 1/26 - 1/31)

= 5.(1 - 1/31)

= 5.30/31

= 150/31

b) B = 4/(11.16) + 4/(16.21) + ... + 4/(61.66)

= 4/5 .[5/(11.16) + 5/(16.21) + ... + 5/(61.66)]

= 4/5.(1/11 - 1/16 + 1/16 - 1/21 + ... + 1/61 - 1/66)

= 4/5.(1/11 - 1/66)

= 4/5 . 5/66

= 2/33

\(D=\frac{5}{18.21}+\frac{5}{21.24}+\frac{5}{24.27}+...+\frac{5}{123.126}\)

\(D=\frac{5}{3}.\left(\frac{3}{18.21}+\frac{3}{21.24}+\frac{3}{24.27}+...+\frac{3}{123.126}\right)\)

\(D=\frac{5}{3}.\left(\frac{1}{18}-\frac{1}{21}+\frac{1}{21}-\frac{1}{24}+\frac{1}{24}-\frac{1}{27}+...+\frac{1}{123}-\frac{1}{126}\right)\)

\(D=\frac{5}{3}.\left(\frac{1}{18}-\frac{1}{126}\right)\)

\(D=\frac{5}{3}.\frac{1}{21}\)

\(\Rightarrow D=\frac{5}{63}\)

\(3D=3\left(\frac{5}{18\cdot21}+\frac{5}{21\cdot24}+...+\frac{5}{123\cdot126}\right)\)

\(=\frac{3\cdot5}{18\cdot21}+...+\frac{3\cdot5}{123\cdot126}\)

\(=5\left(\frac{3}{18\cdot21}+...+\frac{3}{123\cdot126}\right)\)

\(=5\left(\frac{1}{18}-\frac{1}{21}+...+\frac{1}{123}-\frac{1}{126}\right)\)

\(=5\left(\frac{1}{18}-\frac{1}{126}\right)\)

\(=5\left(\frac{7}{126}-\frac{1}{126}\right)\)

\(=5\cdot\frac{6}{126}\)

\(=\frac{30}{126}\)

\(D=\frac{30}{126}\div3=\frac{30}{126}\cdot\frac{1}{3}=\frac{5}{63}\)

                                    \(#Cothanhkhe\)

Bạn trên làm sai rồi

D=5/18.21+5/21.24+…+5/123.126

D.3/5=3/5.(5/18.21+5/21.24+…+5/123.126

D.3/5=3/18.21+3/21.24+…+3/123.126

D.3/5=(1/18-1/21)+(1/21-1/24)+…+(1/123-1/126)

D.3/5=1/18-1/21+1/21-1/24+…+1/123-1/126

D.3/5=1/18-(1/21-1/21)+…+(1/123-1/123)-1/126

D.3/5=1/18+0+0+…+0-1/126

D.3/5=1/18-1/126

D.3/5=7/126-1/126

D.3/5=5/126

D=5/126:3/5

D=5/126.5/3

D=25/378

Vậy D=25/378