Answer
$k = 0.292~N/m$
Work Step by Step
The time to complete half a cycle is 2.60 seconds. Therefore, the period $T$ is 5.20 seconds.
We know that $\omega = \frac{2\pi}{T}$ and $k = m\omega^2$. We can find the force constant $k$.
$k = m\omega^2$
$k = (m)(\frac{2\pi}{T})^2$
$k = (0.200~kg)(\frac{2\pi}{5.20~s})^2$
$k = 0.292~N/m$