19. (12 points)
Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 1 , b _ { 1 } = 0,4 a _ { n + 1 } = 3 a _ { n } - b _ { n } + 4,4 b _ { n + 1 } = 3 b _ { n } - a _ { n } - 4$.
(1) Prove that $\left\{ a _ { n } + b _ { n } \right\}$ is a geometric sequence and $\left\{ a _ { n } - b _ { n } \right\}$ is an arithmetic sequence Therefore, the polar equation of the locus of point $P$ is $\rho = 4 \cos \theta , \theta \in \left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$ .

18. Let $S _ { n }$ denote the sum of the first $n$ terms of $\left\{ a _ { n } \right\}$. Given that $a _ { n } > 0 , a _ { 2 } = 3 a _ { 1 }$, and the sequence $\left\{ \sqrt { S _ { n } } \right\}$ is an arithmetic sequence. Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence.

18. (12 points) Given that all terms of the sequence $\{a_n\}$ are positive numbers, and $S_n$ denotes the sum of the first $n$ terms of $\{a_n\}$. Choose two of the following three statements as conditions and prove the remaining one.
(1) The sequence $\{a_n\}$ is an arithmetic sequence;
(2) The sequence $\{\sqrt{S_n}\}$ is an arithmetic sequence;
(3) $a_2 = 3a_1$.
Note: If different combinations are answered correctly, only the first answer will be scored.

Let the sequence $\{a_n\}$ satisfy $a_1 = 3$, $\frac{a_{n+1}}{n} = \frac{a_n}{n+1} + \frac{1}{n(n+1)}$.
(1) Prove that $\{na_n\}$ is an arithmetic sequence.
(2) Let $f(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$. Find $f'(-2)$.

(15 points) Let the sequence $\{a_n\}$ satisfy $a_1 = 3$, $\frac{a_{n+1}}{n} = \frac{a_n}{n+1} + \frac{1}{n(n+1)}$.
(1) Prove that $\{na_n\}$ is an arithmetic sequence.
(2) Let $f(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$. Find $f'(-2)$.