Question 1: Given x₁=3 and f(x) = 2x + 1, calculate x₂.

Answer: 7

Question 2: If x₁=5 and f(x) = x - 2, find x₃ after applying the iteration twice.

Answer: 1

Question 3: Plot the graph of y=2x for x values from 0 to 5 on the grid.

Answer: Students plot the points (0,0), (1,2), (2,4), (3,6), (4,8), (5,10).

Question 4: Calculate x₄ if x₁=1 and f(x)=x+3, applying the iteration four times.

Answer: 13

Question 5: Construct a triangle on the grid with sides of lengths corresponding to successive x-values if x₁=2 and f(x)=x+2. Draw the first three points.

Answer: Points at (2,2), (4,4), (6,6) with triangle sides accordingly.

Question 6: If x₁=4 and f(x)=3x - 1, what is x₂?

Answer: 11

Question 7: A sequence starts with x₁=1 and follows xₙ₊₁ = 2xₙ + 1. What is x₅?

Answer: 31

Question 8: Identify and correct the mistake: Given x₁=3 and f(x)=x/2, the next term x₂=1.5, then x₃=0.75.

Answer: There is no mistake; the calculations are correct.

Question 9: Solve the multi-step problem: Starting with x₁=2 and f(x)=x+4, find x₃. Then, determine the average of x₁, x₂, and x₃.

Answer: x₂=6, x₃=10, average= (2+6+10)/3=6.

Question 10: Challenge: For f(x)=x² - 3, starting with x₁=2, find x₂ and x₃. Comment on the pattern observed.

Answer: x₂=1, x₃=-2, pattern shows alternating signs and increasing magnitude.

Question 11: Estimate the fixed point of f(x)=x/2 + 1 by solving x = f(x).

Answer: x=2

Question 12: Explain why applying xₙ₊₁ = f(xₙ) repeatedly might lead to convergence or divergence.

Answer: Convergence depends on the function f(x), specifically if it is a contraction mapping; divergence occurs if the values grow without bound or oscillate.