Question

Consider the sequence \[ x_n = 0.5x_{n-1} + 1, n = 1, 2, \ldots \text{ with } x_0 = 0. \text{ Then } \lim_{n \to \infty} x_n \text{ is} \]

• A. 0
• B. 1
• C. 2
• D. \(\infty\)

Hint:

For linear recurrence relations, assume a steady-state solution and solve for the limit.

Answer 1

The Correct Option is C

We are given the recurrence relation \( x_n = 0.5x_{n-1} + 1 \) with the initial condition \( x_0 = 0 \). We need to find \( \lim_{n \to \infty} x_n \). Step 1: Find the steady state solution.
Assume that as \( n \to \infty \), the sequence \( x_n \) approaches a constant value \( L \). Then, the recurrence becomes: \[ L = 0.5L + 1 \] Solving for \( L \): \[ L - 0.5L = 1 $\Rightarrow$ 0.5L = 1 $\Rightarrow$ L = 2 \] Step 2: Conclusion. Therefore, \( \lim_{n \to \infty} x_n = 2 \), so the correct answer is option (C). Final Answer: 2