A thin rod of pure germanium is used to sense temperature…

Step 1. The constant. $\dfrac{E_g}{2k} = \dfrac{0.67\ \text{eV}}{2 \times 8.617 \times 10^{-5}\ \text{eV/K}} = 3888$ K.

Step 2. The resistance at 330 K. The exponent is $3888\ \text{K} \times \left(\dfrac{1}{330\ \text{K}} - \dfrac{1}{300\ \text{K}}\right) = -1.1757$. So $R_2 = 2000\ \Omega \times e^{-1.1757} = 616\ \Omega$.

Step 3. The temperature for $500\ \Omega$. We want $R_2/R_1 = \tfrac{1}{4}$, so the exponent is $\ln\tfrac{1}{4} = -1.3863$. Then $\dfrac{1}{T_2} = \dfrac{1}{300\ \text{K}} - \dfrac{1.3863}{3888\ \text{K}} = 2.9767 \times 10^{-3}\ \text{K}^{-1}$, so $T_2 = 336$ K.

Step 4. The change for each kelvin. $\dfrac{1}{R}\dfrac{dR}{dT} = -\dfrac{E_g}{2kT^2} = -\dfrac{3888\ \text{K}}{(300\ \text{K})^2} = -0.0432\ \text{K}^{-1}$, which is $-4.32\ \%$ for each kelvin.