\(\frac{5}{1.2}+\frac{5}{2.3}+\frac{5}{3.4}+....+\frac{5}{x.\left(x+1\right)}=\frac{44}{9}\)
\(5.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{x.\left(x+1\right)}\right)=\frac{44}{9}\)
\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{x}-\frac{1}{x+1}=\frac{44}{9}:5\)
\(1-\frac{1}{x+1}=\frac{44}{45}\)
\(\frac{1}{x+1}=1-\frac{44}{45}\)
\(\frac{1}{x+1}=\frac{1}{45}\)
=> x + 1 = 45
=> x = 45 - 1
=> x = 44
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