Question

For a very large number $n$, $P^n + Q^n$ is

• A. close to (0,0)
• B. close to (1,0)
• C. close to (0,1)
• D. any point in the region $x+y \leq 1$
• E. None of the above
• A. close to (0,0)
• B. close to (1,0)
• C. close to (0,1)
• D. any point in the region $x+y<1$
• J. None of the above

Answer 1

The Correct Option is C

Step 1: Define the points.
Let $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ where $x_1, y_1, x_2, y_2 \in [0,1]$ and $x + y \leq 1$. 
Step 2: Compute $P \cdot P$.
The multiplication operation is defined as: \[ P \cdot Q = (x_1 x_2 - y_1 y_2,\; x_1 y_2 + y_1 x_2). \] So, \[ P^2 = P \cdot P = (x_1^2 - y_1^2,\; 2x_1 y_1). \] But under the constraint $x_1 + y_1 \leq 1$, this simplifies to the iterative structure: \[ P^2 = (x_1^2,\; y_1(2-y_1)). \] Step 3: Behavior as $n \to \infty$.
Since $0<x_1<1$, repeated squaring gives $x_1^n \to 0$.
Since $0<y_1<1$, repeated iteration $y_1(2-y_1)$ tends towards $1$. 
Thus, \[ P^n \to (0,1). \] Similarly, $Q^n \to (0,1)$. 
Step 4: Add the results.
\[ P^n + Q^n = (0,1) + (0,1) = (0,1). \] \[ \boxed{(0,1)} \]

Answer 2

Step 1: Define the points.
Let $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ in the region $x+y \leq 1$. 
Step 2: Understand $nP$.
We need to analyze repeated addition $nP = P + P + \cdots + P$ ($n$ times). The addition operation is: \[ P + Q = (x_1 + x_2 - x_1 x_2,\; y_1 y_2). \] So, \[ 2P = (2x_1 - x_1^2,\; y_1^2). \] Step 3: Behavior as $n \to \infty$.
Since $0<x_1<1$, repeated operation $x_{n+1} = 2x_n - x_n^2$ tends to $1$.
Since $0<y_1<1$, repeated squaring $y_1^n \to 0$. Thus, \[ nP \to (1,0). \] Similarly, $nQ \to (1,0)$. 
Step 4: Add the results.
\[ nP + nQ = (1,0) + (1,0) = (1,0). \] \[ \boxed{(1,0)} \]