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Ta có : A=20/11×13 + 20/13×15 +20/15×17+...+20/53×55
A = 10 ×( 2/11×13+2/13×15+...12/53×55)
A = 10 ×(1/11-1/13+1/13-1/15+1/15-1/17+...+1/53-1/55)
A = 10 × (1/11-1/55)
A =10 × 4/55
A = 8/11
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= 14/15 x20/21 x 27/28 x ... x 209/210
= 28/30 x 40/42 x 54/56 x ... x 418/420
= 4x7/5x6 x 5x8/6x7 x 6x9/7x8 x ... x 19x22/20x21
= (4x7x5x8x6x9x...x19x22) / (5x6x6x7x7x8x...x20x21)
bạn gộp vào và tối giản đi, cuối cùng đc : 11/15 ( mik ko chắc chắn đâu nên bạn thử lại nhé)
![](https://rs.olm.vn/images/avt/0.png?1311)
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=14/15*20/21*...*209/210
\(=\dfrac{4\cdot7}{5\cdot6}\cdot\dfrac{5\cdot8}{6\cdot7}\cdot...\cdot\dfrac{19\cdot22}{20\cdot21}\)
\(=\dfrac{4\cdot5\cdot6\cdot...\cdot19}{5\cdot6\cdot7\cdot...\cdot20}\cdot\dfrac{7\cdot8\cdot9\cdot...\cdot22}{6\cdot7\cdot8\cdot...\cdot21}=\dfrac{11}{15}\)
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\(\dfrac{1}{15}+\dfrac{1}{21}+\dfrac{1}{28}+...+\dfrac{2}{x\times\left(x+1\right)}=\dfrac{1}{3}\\ \dfrac{2}{30}+\dfrac{2}{42}+\dfrac{2}{56}+...+\dfrac{2}{x\times\left(x+1\right)}=\dfrac{1}{3}\\ 2\times\left(\dfrac{1}{5\times6}+\dfrac{1}{6\times7}+\dfrac{1}{7\times8}+...+\dfrac{1}{x\times\left(x+1\right)}\right)=\dfrac{1}{3}\\ \dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}+...+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{1}{3}:2\\ \dfrac{1}{5}-\dfrac{1}{x+1}=\dfrac{1}{6}\\ \dfrac{1}{x+1}=\dfrac{1}{5}-\dfrac{1}{6}=\dfrac{1}{30}\\ x+1=30\\ x=29\)
Lời giải:
** Sửa đề:
$\frac{1}{15}+\frac{1}{21}+\frac{1}{28}+...+\frac{2}{x(x+1)}=\frac{1}{3}$
$\frac{2}{30}+\frac{2}{42}+\frac{2}{56}+...+\frac{2}{x(x+1)}=\frac{1}{3}$
$\frac{2}{5.6}+\frac{2}{6.7}+\frac{2}{7.8}+...+\frac{2}{x(x+1)}=\frac{1}{3}$
$2(\frac{6-5}{5.6}+\frac{7-6}{6.7}+\frac{8-7}{7.8}+...+\frac{x+1-x}{x(x+1)})=\frac{1}{3}$
$2(\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+...+\frac{1}{x}-\frac{1}{x+1})=\frac{1}{3}$
$2(\frac{1}{5}-\frac{1}{x+1})=\frac{1}{3}$
$\frac{1}{5}-\frac{1}{x+1}=\frac{1}{6}$
$\frac{1}{x+1}=\frac{1}{5}-\frac{1}{6}=\frac{1}{30}$
$\Rightarrow x+1=30$
$\Rightarrow x=29$.