Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.

Problem solving with iteration

Maths

Checking and securing understanding of compound interest calculations

Building on composite functions

Evaluating iterative formulas

Approximating solutions to equations

Signs of a solution

In this unit, pupils continue to explore composite functions, thereby building on their work from the prior unit. Pupils need to understand function notation from the prior unit as this is used in iteration.

In this unit, pupils continue to explore composite functions and develop understanding of iteration, building on their work on functions and proof.

Iteration

The outputs of the formula will eventually converge on 3. 

The iterative formula will not converge. 

Laura has used the equation $$x^2+x-12=0$$ to write the iterative formula $$x_{n+1}=12-(x_n)^2$$.  If 3 is an exact solution, what should happen when she uses $$x_0=3$$?

Which of these could be iterative formulae to find a solution to the equation $$x^2-5x+4=0$$?

Which of these could be iterative formulae to find a solution to the equation $$x^2+2x-6=0$$?

Using the iterative formula $$x_{n+1}=\frac{6}{x_n+2} $$ which values of $$x_0$$ converge on a solution to the equation $$x^2+2x-6=0$$ ?

Which of these iterative formulae converge on the approximate solution $$x=0.459$$ to the equation $$x^2-7x+3=0$$ when $$x_0=2$$?

Using the graph (or otherwise) which values of $$x_0$$ will converge on a solution to the equation $$x^2-7x+3=0$$ using the iterative formula $$x_{n+1}=\frac{(x_n)^2+3}{7}$$?

What are the solutions to the quadratic equation $$x^2-3x-10=0$$?

$$x=1.22$$ and $$x=-2.68$$ to 3 significant figures

$$x=-1.22$$ and $$x=2.68$$ to 3 significant figures

$$x=1.45$$ and $$x=-3.45$$ to 3 significant figures

$$x=-1.45$$ and $$x=3.45$$ to 3 significant figures

What are the approximate solutions to the quadratic equation $$x^2+2x-5=0$$ ?

Which value does the iterative formula $$x_{n+1}=\frac{8}{x_n-2}$$ converge on to 1 significant figure when $$x_0=5$$ ?

Using the iterative formula $$x_{n+1}=\frac{(x_n)^3+1}{2}$$ and $$x_0=0$$ , which of these is a solution to $$x^3-2x+1=0$$ correct to 2 significant figures?

Between which sets of values is there definitely a solution to $$x^2-5x-1=0$$ ?

Which of these are solutions to $$x^2+2x-6=0$$ correct to the stated degree of accuracy?

Problem solving with iteration

Lesson outcome