Iteration

Generate successive approximations and recognise when an iterative sequence converges to a solution.

Curriculum rangeYears 10–11

An iterative formula repeats the same calculation, feeding each answer back in as the next input -- the values gradually settle on a solution.

Starting value

Solving $x^2 - 2x - 3 = 0$ by rearranging for the lone $x$ under the square root:

$x^2 = 2x + 3 \;\Rightarrow\; x = \sqrt{2x + 3}$

$x_{n+1} = \sqrt{2x_n + 3}$

$x_{0}$	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$	$x_{6}$
1	2.236	2.734	2.91	2.97	2.99	2.997

Keep iterating and the values will settle towards a root of the original equation.