A system consisting of two blocks of masses ${\text{m}}_{\text{1}}$  and  ${\text{m}}_{\text{2}}$ connected by a spring of spring constant K. A force  F is applied on  ${\text{m}}_{\text{1}}$ along its weight and removed such that  ${\text{m}}_{\text{2}}$ is just lifted. Then F is

For ${\text{m}}_{\text{1}}$  :  $\text{F}+{\text{m}}_{1}g=k{x}_0$
                       
For ${\text{m}}_2$  :   ${\text{m}}_{\text{2}}$ to lift 
 
                         $kx={\text{m}}_{2}g\to 2$ 
Loss of potential energy  of spring= gain of potential energy of blocks
$$\begin{gathered} \frac{1}{2}k(x_0-x^2)={\text{m}}_{1}g(x_0+x) \\ \frac{1}{2}k(x_0-x)={\text{m}}_{1}g \\ k{x}_0-kx=2{\text{m}}_{1}g \\ \text{F}+{\text{m}}_{1}g-{\text{m}}_{2}g=2{\text{m}}_{1}g \\ \text{F}=({\text{m}}_1+{\text{m}}_2)g \\ \text{F}=({\text{m}}_1+{\text{m}}_2)g \end{gathered}$$