Iteration and Iterative Methods — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers iteration: solving an equation by repeating a process, each answer feeding back in as the next input. By the end of this guide you should be able to apply an iterative formula from a given starting value, produce a sequence of estimates to the accuracy asked for, and state the root that the sequence settles on.
You should also be able to show that an equation can be rearranged into an iterative formula, use the ANS key to carry out the repetition efficiently, recognise convergence and divergence, and explain why the method is needed at all.
The organising idea is that iteration trades exactness for reach. Most equations at GCSE are chosen so that they factorise or fit the quadratic formula. Many equations do not — x³ − 5x + 1 = 0 has perfectly ordinary solutions that no amount of factorising will produce. Iteration gets at those solutions by a different route: start with a rough guess, apply a rule that improves it, and keep going until the improvements stop mattering. The answer is an approximation, but it can be made as accurate as the question requires.
Key terms and definitions
Iteration — repeating a process, with each result becoming the next input.
Iterative formula — a formula in the form xₙ₊₁ = f(xₙ), giving each new estimate from the one before.
x₀ (the starting value) — the first estimate, usually supplied by the question.
xₙ and xₙ₊₁ — the current estimate and the next one. The subscript is just a position counter.
Convergence — the values settling towards a fixed number, which is the root.
Divergence — the values moving further apart instead, so the formula fails to find a root.
Root — a solution of the original equation: a value making it equal zero.
Core concepts
Why the method exists
An equation such as x³ + 3x − 5 = 0 cannot be factorised or solved by formula, yet it clearly has a solution — the expression is negative at x = 1 and positive at x = 2, so it must pass through zero somewhere between.
Iteration finds that value. It will not give an exact surd or fraction, but it will give a decimal to whatever accuracy is asked for, which is usually all the question wants.
Rearranging into an iterative formula
An equation is turned into the form x = something involving x, and that "something" becomes the formula.
Starting from x² − x − 4 = 0: add x + 4 to both sides to get x² = x + 4, then take the square root to give x = √(x + 4). Written as an iterative formula, that is xₙ₊₁ = √(xₙ + 4).
The same equation can be rearranged in several ways, giving different formulas. Some converge quickly, some slowly, and some not at all — which is why exam questions supply the formula rather than leaving the choice to you.
Questions often ask you to show that the rearrangement is correct. Work from the original equation towards the formula, showing each step, rather than asserting the result.
Applying the formula
Substitute x₀ to get x₁, then substitute x₁ to get x₂, and so on. Each output is the next input, and the subscripts simply count how many times the process has run.
Using xₙ₊₁ = √(xₙ + 6) with x₀ = 2:
x₁ = √(2 + 6) = √8 = 2.8284… x₂ = √(2.8284 + 6) = √8.8284 = 2.9713… x₃ = √(2.9713 + 6) = √8.9713 = 2.9952…
The values are closing in on 3, which is the root.
Keep full calculator accuracy between steps. Rounding at each stage lets small errors accumulate, and by the fourth or fifth iteration the answer can be wrong in the digit the question asks for.
Using the ANS key
The repetition is done on the calculator in a few keystrokes rather than by retyping.
Enter the starting value and press equals, so it is held in ANS. Then type the formula with ANS wherever xₙ appears — for instance √(ANS + 6) — and press equals. Each further press of equals produces the next iteration automatically.
This is the intended method, and it is both faster and more accurate than writing intermediate values down.
Convergence, divergence and when to stop
A converging sequence settles: successive values get closer together and home in on a single number.
A diverging sequence does the opposite, with the values growing further apart or shooting off, so no root is found by that formula. Divergence means the rearrangement was a poor one, not that the equation has no solution.
Stop when two consecutive iterations agree to the accuracy the question asks for. If x₃ and x₄ are both 3.1623 to four decimal places, the process has converged and the root is 3.16 to three significant figures.
Give the final answer rounded as instructed, and do not present more digits than the question asks for.
The connection to a graph
The root found is where the graph of the original equation crosses the x-axis, and it is also where the curve y = f(x) meets the line y = x for the rearranged form.
This is why the starting value matters: an equation with more than one root may converge to a different one depending on where you begin, and a question specifying x₀ = 2 rather than x₀ = −1 is steering you to a particular solution.
Worked examples
Example 1: Three iterations
Use xₙ₊₁ = √(3xₙ + 1) with x₀ = 3 to find x₁, x₂ and x₃, each to three decimal places.
x₁ = √(3 × 3 + 1) = √10 = 3.162
x₂ = √(3 × 3.16227… + 1) = √10.4868… = 3.238
x₃ = √(3 × 3.23834… + 1) = √10.7150… = 3.273
The values are rising but by smaller amounts each time, which is convergence. The full unrounded value is carried forward at every stage, even though each answer is written to three decimal places.
Example 2: Showing a rearrangement
Show that x² − x − 4 = 0 can be written as xₙ₊₁ = √(xₙ + 4).
Start from x² − x − 4 = 0.
Add x and 4 to both sides: x² = x + 4.
Take the positive square root of both sides: x = √(x + 4).
Writing this as a repeating process gives xₙ₊₁ = √(xₙ + 4), as required.
Each line follows from the one above, which is what a "show that" question is asking for.
Example 3: Reading off the root
An iteration gives x₃ = 3.16228 and x₄ = 3.16228 to five decimal places. State the root to three significant figures.
Two consecutive iterations now agree to well beyond the accuracy required, so the process has converged.
Rounding to three significant figures gives the root as 3.16.
There is no need to run further iterations once successive values agree to the accuracy the question asks for.
Common mistakes and how to avoid them
Rounding between iterations. Carry the full calculator value forward. Round only the values you write down, and only at the end for the final answer.
Using x₀ twice. Each iteration uses the result of the one before, not the original starting value.
Miscounting the subscripts. x₀ is the starting value, so the first calculation produces x₁. A question asking for x₃ wants three calculations, not four.
Retyping instead of using ANS. The ANS key removes both the effort and most of the rounding risk.
Assuming divergence means no solution. It means that particular rearrangement does not work from that starting value.
Giving too many digits. Round to the accuracy stated, and say what the root is rather than leaving a list of iterations.
Asserting a rearrangement rather than showing it. A "show that" question needs the algebraic steps written out.
Exam technique for "Iteration and Iterative Methods"
Write x₀, x₁, x₂ and so on down the page as a labelled list. The marks are for the correct values in the correct positions, and a labelled list makes both visible.
Use the ANS key and say that you have. It is the intended method and keeps full accuracy automatically.
Record each iteration to more decimal places than the final answer needs, so the rounding at the end is safe.
For a "show that" rearrangement, start from the given equation and work forwards line by line to the formula.
State the root explicitly at the end, rounded as instructed, rather than stopping at the last iteration.
Check the subscript the question asks for before deciding how many times to press equals.
Quick revision summary
Iteration solves equations that cannot be factorised, by repeating a process: each answer becomes the next input.
An iterative formula has the form xₙ₊₁ = f(xₙ), obtained by rearranging the equation to make x the subject in a repeatable way. So x² − x − 4 = 0 becomes xₙ₊₁ = √(xₙ + 4).
Start from x₀, which the question supplies, and work forwards. x₀ is the starting value, so the first calculation gives x₁.
Use the ANS key: enter the starting value, type the formula with ANS in place of xₙ, then press equals repeatedly.
Keep full accuracy between steps. Rounding early makes the later iterations wrong in exactly the digit being asked for.
The values converge when they settle towards a single number, which is the root; they diverge when they spread apart, meaning that rearrangement fails from that start.
Stop when two consecutive iterations agree to the required accuracy, then state the root rounded as instructed.