(i) By the binomial theorem $(\sqrt{2} + 1)^{10} = \sum_{i=0}^{10} C_i (\sqrt{2})^i$, where $C_i$ are appropriate constants. Write the value of $i$ for which $C_i (\sqrt{2})^i$ is the largest among the 11 terms in this sum.
(ii) For every natural number $n$, let $(\sqrt{2} + 1)^n = p_n + \sqrt{2} q_n$, where $p_n$ and $q_n$ are integers. Calculate $\lim_{n \rightarrow \infty} \left(\frac{p_n}{q_n}\right)^{10}$.

For the sequence $\left\{ \left( \frac { 2 x - 1 } { 4 } \right) ^ { n } \right\}$ to converge, let $k$ be the number of integers $x$. Find the value of $10 k$. [3 points]

When $\lim _ { n \rightarrow \infty } \frac { a \times 6 ^ { n + 1 } - 5 ^ { n } } { 6 ^ { n } + 5 ^ { n } } = 4$, what is the value of the constant $a$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 4 } { 3 }$
(5) $\frac { 3 } { 2 }$

Find the value of $\lim _ { n \rightarrow \infty } \frac { 3 \times 9 ^ { n } - 13 } { 9 ^ { n } }$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 1 and common ratio $r$ ($r > 1$), let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. When $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { S _ { n } } = \frac { 3 } { 4 }$, find the value of $r$. [3 points]

12. For $x \in R$, $\operatorname { limn } \rightarrow \infty ( ( x - 3 ) / ( x + 2 ) ) x =$
(A) e
(B) $e - 1$
(C) $e - 5$
(D) e 5

For each positive integer $n$, let $$y _ { n } = \frac { 1 } { n } ( ( n + 1 ) ( n + 2 ) \cdots ( n + n ) ) ^ { \frac { 1 } { n } }$$ For $x \in \mathbb { R }$, let $[ x ]$ be the greatest integer less than or equal to $x$. If $\lim _ { n \rightarrow \infty } y _ { n } = L$, then the value of $[ L ]$ is $\_\_\_\_$.