Let us assume the mass of the neutron is $m_{\mathrm{n}}=m$ and mass of the nucleus is $m_{\mathrm{A}}=Am$

Also, let us assume that the initial and the final velocities of the neutron are $v_{\mathrm{i}}$ and $v_{\mathrm{f}}$ respectively. 

Using the formula of final velocity for a perfectly elastic collision we get,

$v_{\mathrm{f}}=\frac{\left(m-Am\right)}{(m+Am)}{v}_{\mathrm{i}}=\left(\frac{1-A}{1+A}\right)v_{\mathrm{i}}$

$\Rightarrow \frac{v_{\mathrm{f}}}{v_{\mathrm{i}}}=\left(\frac{1-A}{1+A}\right)$

$\Rightarrow \frac{K_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}{K_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}={\left(\frac{1-A}{1+A}\right)}^2$