Question 1: Calculate the approximate value of x in the equation x^2 = 20 using iterative approximation methods. Show your working.

Answer: 4.47

Question 2: Using successive approximation, estimate the cube root of 50 to two decimal places. Show your steps.

Answer: 3.68

Question 3: Construct an iterative sequence to approximate the square root of 81 starting from 5. Show the first three iterations.

Answer: ["Iteration 1: (5 + 81/5)/2 = 9.5","Iteration 2: (9.5 + 81/9.5)/2 ≈ 9.0","Iteration 3: (9 + 81/9)/2 = 9.0"]

Question 4: Fluency & Practice: Find the approximate solution to x^3 = 100 using iteration, starting from x=4. Show your working.

Answer: 4.64

Question 5: Problem Solving & Reasoning: A car's speed is estimated by iterating the average of its current speed and a target speed of 60 km/h. Starting from 20 km/h, perform two iterations and explain the process.

Answer: First iteration: (20 + 60)/2 = 40 km/h; second iteration: (40 + 60)/2 = 50 km/h. The process involves averaging current speed with target to approach 60 km/h gradually.

Question 6: Estimate the value of x in the equation 3^x = 50 using iterative approximation starting from x=3.5. Show your steps.

Answer: 3.88

Question 7: Real-world Application: A cyclist aims to reach a target distance of 100 km by increasing their daily distance gradually. Starting with 20 km on the first day and increasing by half the previous day's distance each day, how many days will it take to reach or exceed 100 km? Explain your reasoning.

Answer: This is a geometric sequence: Day 1: 20 km, Day 2: 30 km, Day 3: 45 km, Day 4: 67.5 km, Day 5: 101.25 km. It takes 5 days.

Question 8: Construct an iterative process to approximate the fourth root of 16, starting from an initial guess of 2. Show the first three iterations.

Answer: ["Iteration 1: (1/4)(2 + 16/ (2^3)) = 2","Iteration 2: (1/4)(2 + 16/ (2^3)) = 2","Iteration 3: same as above, converges at 2"]

Question 9: Challenge & Extension: Use iteration to approximate the solution to x^2 + x - 6 = 0, starting with x=2. Show your first two iterations.

Answer: ["Iteration 1: x = (6 - x^2)/x, plug x=2: (6 - 4)/2=1","Iteration 2: x = (6 - 1)/1=5"]

Question 10: Mixed Review: A sequence is defined by x_{n+1} = (x_n + 81/x_n)/2, starting from x_0=10. Find x_2 and comment on the convergence.

Answer: ["x_1: (10 + 81/10)/2 = (10 + 8.1)/2 = 9.05","x_2: (9.05 + 81/9.05)/2 ≈ (9.05 + 8.95)/2 = 9.0","The sequence converges to 9, which is √81."]

Question 11: Error Analysis: A student incorrectly completes an iterative step for approximating √50 as 7.2 instead of the correct value. Identify the common mistake and explain how to fix it.

Answer: The student might have misapplied the formula or performed a calculation error. To fix it, they should carefully follow the iterative formula: x_{n+1} = (x_n + 50/x_n)/2, ensuring correct substitution and calculation.