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.

Approximating solutions to equations

Maths

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.

Iteration

Which value does the iterative formula $$x_{n+1}=\frac{5}{x_n}-3$$ converge on to 1 significant figure when $$x_0=2$$ ?

Which value does the iterative formula $$x_{n+1}=\frac{5}{x_n}-3$$ converge on to 2 significant figures when $$x_0=2$$ ?

The iterations alternate between positive and negative.

The difference between iterations is decreasing. 

How can we tell if an iterative formula is converging for a given value of $$x_0$$?

Which of these could be an iterative formula to find a solution to $$x^3-3x-5=0$$ ?

To find a solution to the equation $$x^3+6x-10=0$$ the iterative formula $$x_{n+1}=\frac{10-(x_n)^3}{6}$$ can be used. Using $$x_0=1$$ find the value of $$x_3$$ 

Using the iterative formula $$x_{n+1}=\frac{10-(x_n)^3}{6}$$ and $$x_0=1$$ there is a solution to $$x^3+6x-10=0$$ correct to 2 significant figures when $$x=$${{}} 

What is 0.0647399 rounded to 2 significant figures?

What is the value of the expression $$x^3 +3x-12$$ when $$x=2$$?

For which values of $$x$$ are we able to evalate the expression $$\sqrt{6-2x}$$?

Use the iterative formula $$P_{t+1}=2.5(P_t-50)$$ with $$P_0=200$$ to find the value of $$P_4$$. Write down your full calculator display. 

Using the iterative formula $$x_{n+1}=10-4(x_n)$$ with $$x_0=1$$ which of these is the value of $$x_8$$?

Using the iterative formula $$x_{n+1}=\sqrt{10-2(x_n)}$$ and $$x_0=3$$ match up the iterations with their values. 

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