Question 1: Given that x₀ = 3 and the iteration formula xₙ₊₁ = 2xₙ + 1, calculate the value of x₁ and x₂.

Answer: x₁=7, x₂=15

Question 2: If x₀ = 5 and the iteration is xₙ₊₁ = xₙ/2, what is x₃?

Answer: x₃=1.25

Question 3: Construct the first three steps of the iteration starting from x₀=4 with xₙ₊₁ = 3xₙ - 2.

Answer: x₁=10, x₂=28, x₃=82

Question 4: A student mistakenly uses the formula xₙ₊₁ = 2xₙ + 1 but forgets to substitute the initial value when calculating x₁. If x₀=2, what incorrect value might they record, and what is the correct value?

Answer: Incorrect: 5; Correct: 5

Question 5: Plot the graph of y = 0.5x + 2 for x values from -4 to 4.

Answer: Students will draw a straight line based on the points (-4, 0), (0, 2), (4, 4).

Question 6: Explain why repeatedly applying the iteration xₙ₊₁ = 0.9xₙ will lead to the sequence approaching a specific value. What is that value?

Answer: It approaches 0 because 0.9<1, so the sequence converges to 0.

Question 7: Suppose a student incorrectly computes x₁ = 2x₀ + 1 as 2x₀ - 1. Given x₀=3, what is the mistake, and what should the correct calculation be?

Answer: Mistake: subtraction instead of addition; Correct: 2×3 + 1 = 7

Question 8: Given the iteration xₙ₊₁ = xₙ² - 1 starting with x₀=2, find x₁ and x₂. Identify any potential mistake if the student forgets to square xₙ.

Answer: x₁=3, x₂=8

Question 9: A student claims that for xₙ₊₁= 1.5xₙ + 0.5, the sequence will always increase if x₀ > 1. Verify or refute this statement with calculations.

Answer: Refute; for x₀=2, x₁=4, x₂=7, sequence increases; for x₀=0.5, x₁=1.25, sequence increases. So, it increases if x₀ > 1, but decreases if x₀<1.

Question 10: Explain a common misconception students might have when applying the iteration formula xₙ₊₁ = 2xₙ + 1, especially regarding initial values.

Answer: Students may forget to correctly substitute initial values or think the sequence is always increasing without checking the actual calculations.

Question 11: Challenge: Starting with x₀=1, iterate xₙ₊₁ = xₙ² - 2 for four steps. What do you observe about the sequence? Show your calculations.

Answer: x₁= -1, x₂= -1, x₃= -1, x₄= -1; sequence stabilizes at -1 after the first step due to squaring and subtraction.