If ${x^2}^r$ Occurs in ${\left(x+\frac{{\displaystyle 2}}{{\displaystyle x^2}}\right)}^n$, then $n-2r$must be of the form

The general term of  ${\left(x+\frac{{\displaystyle 2}}{{\displaystyle x^2}}\right)}^n$ is

$T_{R+1}{=}^n{C}_R{\left(x\right)}^{n-R}{\left(\frac{{\displaystyle 2}}{{\displaystyle x^2}}\right)}^R$

${=}^n{C}_R{x}^{n-3R}{2}^R$

For $x^{2r}$ occurs, it means

$n-3R=2r\Rightarrow n-2r=3R$

Hence, $n-2r$ is of the form $3k$.