Ta có \(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+....+\frac{2}{x\left(x+1\right)}\)
=\(2\left(\frac{1}{42}+\frac{1}{56}+....+\frac{1}{x\left(x+1\right)}\right)\)
=\(2\left(\frac{1}{6.7}+\frac{1}{7.8}+.....+\frac{1}{x\left(x+1\right)}\right)\)
=\(2\left(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+....+\frac{1}{x}-\frac{1}{x+1}\right)\)
=\(2\left(\frac{1}{6}-\frac{1}{x+1}\right)\)
=>\(2\left(\frac{1}{6}-\frac{1}{x+1}\right)=\frac{2}{9}\)
=>\(\frac{1}{6}-\frac{1}{x+1}=\frac{2}{9}:2\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{6}-\frac{1}{9}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{18}\)
=> x+1=18
x=18-1
x=17
2 / 6,7 + 2 / 7.8 + ... + 2 / x (x + 1) = 2/9 Ta có: 1 / x - 1 / (x + 1) = (x + 1 - x) / x (x 1) = 1 / x (x + 1) ⇒ 2 / x (x + 1) = 2 / x - 2 / (x + 1) (*) trở thành: 2/6 - 2/7 + 2/7 - 2/8 + ... + 2 / x - 2 / (x + 1) = 2/9 ⇔ 2/6 - 2 / (x + 1) = 2/9 ⇔ 1/6 - 1 / (x + 1) = 1/9 ⇔ 1 / (x + 1) = 1/18 ⇔ x + 1 = 18 ⇔ x = 17 Đáp số: x = 17