Question 1: Calculate the value of x after 5 iterations of the formula x_{n+1} = 0.5(x_n + 3/x_n), starting with x_0 = 2.

Answer: 2.79

Question 2: Using the iterative formula x_{n+1} = 0.8x_n + 1, find an approximate value of x after 4 iterations starting with x_0 = 0.

Answer: 3.66

Question 3: Construct a sequence using the rule x_{n+1} = 1.2x_n - 2, starting from x_0 = 4. Calculate x_3.

Answer: 0.96

Question 4: Plot the graph of y = 2x and identify the approximate value of x when y = 10.

Answer: 5

Question 5: A calculator uses iterative approximation to find the square root of 50. If the initial guess is 7, perform two iterations of the formula x_{n+1} = 0.5(x_n + 50/x_n). What is the approximate square root after two iterations?

Answer: 7.22

Question 6: Construct a triangle on the grid with sides approximately 3, 4, and 5 units to illustrate a Pythagorean triple. Use iteration to approximate the hypotenuse length starting from sides 3 and 4.

Answer: 5

Question 7: Error Analysis: A student performs 3 iterations of x_{n+1} = 0.6x_n + 2 starting from x_0=1 and concludes the solution is 2.8. Identify the mistake and perform the correct 3rd iteration.

Answer: 2.52

Question 8: Using iterative methods, estimate the cube root of 27 starting from an initial guess of 3. Perform two iterations of the formula x_{n+1} = (2/3)x_n + 27/(3x_n^2).

Answer: 3

Question 9: Solve the equation x^2 - 7x + 10 = 0 using iterative approximation starting with x_0=2. Use the iteration x_{n+1} = (1/2)(x_n + 10/x_n). Find x_3.

Answer: 3.16

Question 10: A real-world problem: A car accelerates according to the formula v_{n+1} = 0.9v_n + 5, where v_0=10 km/h. Find an approximate velocity after 4 iterations.

Answer: 18.95