Mastering Iteration for GCSE Maths: A Step-by-Step Guide

Iteration for GCSE and A Level

 

Aim: To use iteration to find an approximate solution to the equation x² - x - 5 = 0.

Step 1: Rearrange the equation to get the iterative form, x² = x + 5..

Step 2: Divide both sides of the equation by x to get x = 1 + 5/x - the iterative form.

Step 3: Now write it in this form: x_n+1 = 1 + 5/x_n

Step 4: Use an initial value to trigger off iteration such as x₀ = 1.

Step 5: We can now use the iterative formula. We can start with an initial guess of x₀=1 and then use the iterative formula to repeat, until we get a value that becomes relatively constant.

Step 6: Approximate the value to the required decimal places and it is a root of the quadratic equation.

x(0) = 1

x₁ = 6.0 

x₂ = 1.83333333333 

x₃ = 3.72727272727

x₄ = 2.34146341463 

x₅ = 3.13541666667 

x₆ = 2.59468438538

x₇ = 2.92701664533

x₈ = 2.708223972

x₉ = 2.84622839606

x₁₀ = 2.75671074286 

x₁₁ = 2.81375576416 

x₁₂ = 2.77698436505

x₁₃ = 2.80051427834

x₁₄) = 2.78538636231

x₁₅ = 2.79508310504

x₁₆ = 2.78885557677

x₁₇ = 2.79285010011

x₁₈ = 2.79028584449

x₁₉ = 2.79193110623

x₂₀ = 2.79087513616

As you can see, the solution/root is getting closer to 2.79. Therefore, the answer is x = 2.79 to 2 d.p.

You can practise the above interactively with the following Python programme: