Question 1: Given xₙ₊₁ = 2xₙ + 1 and starting with x₀ = 3, calculate the value of x₁ and x₂.

Answer: x₁=7, x₂=15

Question 2: Plot the graph of y = 3x + 2 for x values from -3 to 3.

Answer: Students will draw the line on grid

Question 3: Construct a triangle with vertices at (0,0), (x,y), and (x+2,y+3). Use the grid to illustrate.

Answer: Students will sketch on grid

Question 4: Starting with x₀ = 4, find xₙ₊₁ for n=0, given the function xₙ₊₁ = -xₙ + 6. Show your work.

Answer: x₁=2

Question 5: If xₙ₊₁ = 0.5xₙ + 1 and x₀ = 8, compute x₁, x₂, and x₃.

Answer: x₁=5, x₂=3.5, x₃=2.75

Question 6: Explain why the sequence defined by xₙ₊₁ = 0.9xₙ + 0.1 tends to a fixed point as n becomes large, starting from any initial x₀.

Answer: Students should describe convergence due to the coefficient less than 1, leading to a fixed point at x = 1

Question 7: Solve for x if the fixed point of xₙ₊₁ = 3xₙ + 4 is x.

Answer: x = -2

Question 8: A sequence is defined by xₙ₊₁ = 0.7xₙ + 2. Starting with x₀=10, find the value of x₄.

Answer: x₄ ≈ 3.66

Question 9: Identify the error in the following iterative calculation: starting with x₀=5, using xₙ₊₁ = 2xₙ - 3, the student says x₁=7. Is this correct? Why or why not?

Answer: No, x₁=2*5-3=7, so it is correct. The student is correct if they computed accordingly.

Question 10: Challenge: Derive a general formula for xₙ in terms of x₀ for the recurrence xₙ₊₁ = 2xₙ + 3, and verify it for n=3 with x₀=1.

Answer: xₙ=2ⁿx₀ + 3(2ⁿ - 1)