Tìm giá trị của x: 1/21 + 1/28 + 1/36 + ... + 2/(x(x+1)) = 2/9

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

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

\[2\left[ {\frac{1}{{6 \cdot 7}} + \frac{1}{{7 \cdot 8}} + \frac{1}{{8 \cdot 9}} + ... + \frac{1}{{x\left( {x + 1} \right)}}} \right] = \frac{2}{9}\]

\[\frac{1}{6} - \frac{1}{7} + \frac{1}{7} - \frac{1}{8} + \frac{1}{8} - \frac{1}{9} + ... + \frac{1}{x} - \frac{1}{{x + 1}} = \frac{2}{9}:2 = \frac{1}{9}\]

\[\frac{1}{6} - \frac{1}{{x + 1}} = \frac{1}{9}\]

\[\frac{1}{{x + 1}} = \frac{1}{6} - \frac{1}{9} = \frac{1}{{18}}\]

x + 1 = 18

x = 18 ‒ 1 = 17.

Vậy x = 17.