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 $\rho = 4 \cos \theta , \theta \in \left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$ .
Solution:
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 $\rho = 4 \cos \theta , \theta \in \left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$ .