Đặt \(A=\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{x\left(x+1\right)}\)

=> \(A=\frac{2}{6.7}+\frac{2}{7.8}+\frac{2}{8.9}+...+\frac{2}{x\left(x+1\right)}\)

\(\frac{A}{2}=\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+...+\frac{1}{x}-\frac{1}{x+1}\)

=> \(\frac{A}{2}=\frac{1}{6}-\frac{1}{x+1}=\frac{x+1-6}{6\left(x+1\right)}=\frac{x-5}{6\left(x+1\right)}\) => \(A=\frac{x-5}{3\left(x+1\right)}=\frac{2}{9}\)

<=> 3(x-5)=2(x+1)  <=> 3x-15=2x+2  <=> x=17

Đáp số: x=17

a/(x+1)+(x+2)+...+(x+100)=5750

x+1+x+2+...+x+100=5750

100x+(1+2+...+100)=5750

100x+50.(100+1)=5750

100x+5050=5750

100x=700

x=7.

b/ 1/1.2+1/2.3+...+1/x(x+1)=2015/2016

1/1-1/2+1/3-1/4+...+1/x-1/x+1=2015/2016

1-1/x+1=2015/2016

1/x+1=1/2016

x+1=2016

x=2015

\(\left[1-\frac{1}{21}\right]\times\left[1-\frac{1}{28}\right]\times\left[1-\frac{1}{36}\right]\times...\times\left[1-\frac{1}{1326}\right]\)

\(=\frac{20}{21}\times\frac{27}{28}\times\frac{35}{36}\times...\times\frac{1325}{1326}\)

\(=\frac{40}{42}\times\frac{54}{56}\times\frac{70}{72}\times...\times\frac{2650}{2652}\)

\(=\frac{5\times8}{6\times7}\times\frac{6\times9}{7\times8}\times\frac{7\times10}{8\times9}\times...\times\frac{50\times53}{51\times52}\)

\(=\frac{5\times6\times7\times...\times50}{6\times7\times8\times...\times51}\times\frac{8\times9\times10\times...\times53}{7\times8\times9\times...\times52}\)

\(=\frac{5}{51}\times\frac{53}{7}\)

\(=\frac{265}{357}\)

= 20/21 . 27/28 . 35/36 . ...... 1325/1326

= 2/2(20/21 . 27/28 . 35/36 . ...... 1325/1326)

= 40/42. 54/56 . 70/72 ......2650/2652

= 5.8 / 6.7 . 6.9/ 7.8 . 7.10/8.9 ..... 50.53/51.52

.......Sau đọc t cũng k hiểu nữa

Nguồn: của bn Thành :>>>>>

* ĐK: \(x\ne0\)

Đề ra ...<=> \(\frac{2}{42}+\frac{2}{56}+\frac{2}{72}+...+\frac{1}{x\left(x+1\right)}=\frac{2}{9}\)

<=> \(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{2}{x\left(x+1\right)}=\frac{1}{9}\)

<=> \(\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{x\left(x+1\right)}+\frac{1}{x\left(x+1\right)}=\frac{1}{9}\)

<=>\(\frac{1}{6}-\frac{1}{x+1}+\frac{1}{x\left(x+1\right)}=\frac{1}{9}\)

<=>\(\frac{1}{x+1}\left(1-\frac{1}{x}\right)=\frac{1}{6}-\frac{1}{9}\)

<=> \(\frac{x-1}{x\left(x+1\right)}=\frac{1}{36}\)

<=> \(\frac{x-1}{x\left(x-1\right)}=\frac{x-1}{36.\left(x-1\right)}\)

=> x(x-1) = 36. (x-1) => x =36

\(\frac{2}{2}.\left(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{x+\left(x+1\right)}\right)=\frac{2}{9}\)

\(2\left(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{1}{x.\left(x+1\right)}\right)=\frac{2}{9}\)

\(\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{x.\left(x+1\right)}=\frac{1}{9}\)

\(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{1}{9}\)

\(\frac{1}{6}-\frac{1}{x+1}=\frac{1}{9}\)

\(\frac{1}{x+1}=\frac{1}{6}-\frac{1}{9}\)

\(\frac{1}{x+1}=\frac{1}{18}\)

x+1=18

x=18-1

x=17

1/21 + 1/28 + 1/36 + ... + 2/x(x + 1) = 2/9

1/2 × (1/21 + 1/28 + 1/36 + ... + 2/x(x + 1) = 1/2 × 2/9

1/42 + 1/56 + 1/72 + ... + 1/x(x + 1) = 1/9

1/6×7 + 1/7×8 + 1/8×9 + ... + 1/x(x + 1) = 1/9

1/6 - 1/7 + 1/7 - 1/8 + 1/8 - 1/9 + ... + 1/x - 1/x + 1 = 1/9

1/6 - 1/x + 1 = 1/9

1/x + 1 = 1/6 - 1/9

1/x + 1 = 3/18 - 2/18

1/x + 1 = 1/18

=> x + 1 = 18

=> x = 18 - 1

=> x = 17