6. (8 points) By expanding the expression $(1+\sqrt{7})^{211}$ using the binomial theorem, we obtain terms of the form $C_{211}^{k}(\sqrt{7})^{k}$. Find the value of $k$ for which such a term attains its maximum value.
Answer: 153

Solution: The ratio of two consecutive terms $\frac{C_{211}^{k+1}(\sqrt{7})^{k+1}}{C_{211}^{k}(\sqrt{7})^{k}}$ is greater than 1 when $k<$ $\frac{211 \sqrt{7}-1}{\sqrt{7}+1}$. Then the terms increase up to $\left[\frac{211 \sqrt{7}-1}{\sqrt{7}+1}\right]+1$, and then decrease.