解析这道题---**Question 19**
**Question Stem:**
Given the hyperbola C: x² - y² = m (m > 0), point P₁(5,4) is on C. k is a constant, 0 < k < 1. Points Pₙ (n=2, 3, ...) are constructed sequentially as follows: A line with slope k is drawn through P_{n-1} which intersects the left branch of C at point a_{n-1}. Let Pₙ be the point symmetric to a_{n-1} with respect to the y-axis. The coordinates of Pₙ are denoted as (xₙ, yₙ).
**(1) Sub-question:** If k = 1/2, find x₂, y₂.
**(2) Sub-question:** Prove that the sequence {xₙ - yₙ} is a geometric progression with a common ratio of (1+k)/(1-k).
**(3) Sub-question:** Let Sₙ be the area of triangle △PₙP_{n+1}P_{n+2}. Prove that for any positive integer n, Sₙ = S_{n+1}.