Question 1: Starting from x=1, perform two iterations of the formula x_{n+1} = 0.5x_n + 2. Calculate the value after the second iteration.

Answer: 3

Question 2: Use the iterative formula y_{n+1} = 0.8y_n + 1 to find the approximate value of y after 4 iterations starting from y=0.

Answer: 2.83

Question 3: Construct two iterations of the formula x_{n+1} = 0.3x_n + 4 starting from x=10. Record the value after each iteration.

Answer: Iteration 1: 5.2; Iteration 2: 5.56

Question 4: A calculator uses the iterative method x_{n+1} = √(x_n + 3) to approximate √7, starting from x=2. Perform 3 iterations and give your final answer.

Answer: 2.65

Question 5: Solve the problem: Starting with x=0, perform 3 iterations of x_{n+1} = 0.6x_n + 1. What is the value of x after these iterations?

Answer: 2.2

Question 6: Explain why the iterative process x_{n+1} = 0.1x_n + 10 converges or diverges. Support your answer with calculations.

Answer: Diverges because the multiplier 0.1 < 1, but the constant term is large; the sequence approaches 10 but never stabilizes if starting far from 10.

Question 7: Given the iterative formula y_{n+1} = 0.5y_n + 3, find the fixed point and explain its significance.

Answer: Fixed point at y=6; it is the value y approaches as n becomes large.

Question 8: A real-world scenario: A bacteria culture size is modelled by the iterative formula N_{n+1} = 0.9N_n + 50, starting with N=100. Calculate the population after 3 iterations.

Answer: 119.29

Question 9: Identify the error in this iterative calculation: Starting with x=2, the calculation proceeds with x_{n+1} = 2x_n + 1, and after 3 iterations, the student reports x=15. Correct the mistake.

Answer: Incorrect: each iteration is 2x + 1; starting from 2: 1st: 5, 2nd: 11, 3rd: 23.

Question 10: Challenge: Find the fixed point of the iterative process x_{n+1} = 1.2x_n - 4 and determine whether the sequence converges or diverges.

Answer: Fixed point at x=20; diverges because 1.2 > 1.