The equilibrium constant of an endothermic reaction is $2.0$ at $400\mathrm{K}$ and $8.0$ at $500\mathrm{K}$. Assuming $\Delta H^{\circ }$ is constant, the value of $K$ at $450\mathrm{K}$ is closest to

$\ln K$ is linear in $1/T$. The fraction of the way from $400$ to $450\mathrm{K}$ on the $1/T$ scale is $(1/400-1/450)/(1/400-1/500)=0.5556$. Therefore $\ln K_{450}=\ln 2.0+0.5556\ln 4.0=1.4633$ and $K_{450}=4.32$. Interpolating linearly in $T$ gives 5.0 and taking the geometric mean gives 4.0, both wrong.