6. (8 points) By expanding the expression $(1+\sqrt{7})^{205}$ using the binomial theorem, we obtain terms of the form $C_{205}^{k}(\sqrt{7})^{k}$. Find the value of $k$ for which such a term attains its maximum value.
Answer: 149

Solution: The ratio of two consecutive terms $\frac{C_{205}^{k+1}(\sqrt{7})^{k+1}}{C_{205}^{k}(\sqrt{7})^{k}}$ is greater than 1 when $k<$ $\frac{205 \sqrt{7}-1}{\sqrt{7}+1}$. Then the terms increase up to $\left[\frac{205 \sqrt{7}-1}{\sqrt{7}+1}\right]+1$, and then decrease.