6. (8 points) By expanding the expression $(1+\sqrt{5})^{213}$ using the binomial theorem, we obtain terms of the form $C_{213}^{k}(\sqrt{5})^{k}$. Find the value of $k$ for which such a term attains its maximum value.
Answer: 147

Solution: The ratio of two consecutive terms $\frac{C_{213}^{k+1}(\sqrt{5})^{k+1}}{C_{213}^{k}(\sqrt{5})^{k}}$ is greater than 1 when $k<$ $\frac{213 \sqrt{5}-1}{\sqrt{5}+1}$. Then the terms increase up to $\left[\frac{213 \sqrt{5}-1}{\sqrt{5}+1}\right]+1$, and then decrease.