Question 1: Given xₙ₊₁ = 2xₙ + 3, if x₀ = 1, find x₁, x₂, and x₃.

Answer: x₁=5; x₂=13; x₃=29

Question 2: Calculate the value of x after 4 iterations starting from x₀=2, where xₙ₊₁ = xₙ² - 1.

Answer: 15,625

Question 3: Plot the graph of y=0.5x + 2 on the grid. What is the fixed point where x = y?

Answer: x=4

Question 4: Construct a triangle with sides length 3, 4, and 5 on the grid. Use the iteration formula xₙ₊₁ = xₙ + 2 to determine the length of the side after 3 iterations starting from 3.

Answer: 7

Question 5: Using the function xₙ₊₁ = 3 - 0.5xₙ, find the fixed point of the iteration.

Answer: x=2

Question 6: Determine whether the sequence xₙ₊₁ = 0.8xₙ + 1 converges when starting from x₀=0. Show your reasoning.

Answer: Yes, it converges to 5, since fixed point x=5; 0.8<1 ensures convergence.

Question 7: Solve for x: x = f(x), where f(x) = x² - 3x + 2. List all fixed points.

Answer: x=1 or x=2

Question 8: A population model is given by xₙ₊₁ = 0.9xₙ + 10. Starting from x₀=0, estimate the population after 5 iterations.

Answer: Approximately 53.4

Question 9: Identify the error in the following statement: 'Since xₙ₊₁ = 2xₙ + 1, starting from x₀=0, the sequence will diverge.'

Answer: Incorrect; the sequence diverges because 2>1, leading to divergence, but the student's reasoning may be flawed if they assume convergence.

Question 10: Determine the fixed point of the iteration xₙ₊₁ = xₙ/2 + 3. Is it stable?

Answer: Fixed point: x=6; it is stable since |f'(x)|=0.5<1.