1/2+1/6+1/12+1/20+....+1/x(x+1)=2022/2023

`1/2 + 1/6 + 1/12 + 1/20 + .... +` $\frac{1}{x.(x+1)}$ `= 2022/2023`

$\frac{1}{1.2}$ + $\frac{1}{2.3}$  + $\frac{1}{3.4}$  + $\frac{1}{4.5}$  `+ .... + = 2022/2023`

`1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + .... + 1/x -` $\frac{1}{x + 1}$ `= 2022/2023`

`1/1 -` $\frac{1}{x + 1}$ `= 2022/2023`

$\frac{1}{x + 1}$ `= 1/1 - 2022/2023`

$\frac{1}{x + 1}$ `= 1/2023`

`⇒ x + 1 = 2023`

    `x = 2023 - 1`

    `x = 2022`