A man starts pulling on the cord when the spring is in its...

The equilibrium position of the block is reached when the net force acting on it is zero. At this position, the constant applied force $F$ is balanced by the restoring force of the spring.

$F = kx_0 \Rightarrow x_0 = \dfrac{F}{k}$

According to the work-energy theorem, the net work done on the block equals its change in kinetic energy:

$W_{\text{net}} = \Delta K$

The work done by the constant force $F$ is:

$W_F = F x_0 = F \left(\dfrac{F}{k}\right) = \dfrac{F^2}{k}$

The work done by the spring force is:

$W_s = -\dfrac{1}{2} k x_0^2 = -\dfrac{1}{2} k \left(\dfrac{F}{k}\right)^2 = -\dfrac{F^2}{2k}$

Since the block starts from rest, its initial kinetic energy is zero. The kinetic energy $K$ at the equilibrium position is:

$K = W_F + W_s = \dfrac{F^2}{k} - \dfrac{F^2}{2k} = \dfrac{F^2}{2k}$