What you'll learn
This topic covers iteration — using a repeating process to find approximate solutions to equations that cannot be solved exactly. In this guide you will learn what an iterative formula is, how to apply it step by step, how to use the answer (ANS) button efficiently, how to spot convergence, and how iteration relates to a graph. This is a higher-tier algebra skill.
Key terms and definitions
Iteration — repeating a process, each result feeding into the next.
Iterative formula — a formula of the form xₙ₊₁ = f(xₙ).
Starting value (x₀) — the first estimate you put in.
Convergence — when the values settle towards a fixed solution.
Root — a solution of the equation (where it equals zero).
Core concepts
What iteration does
Iteration solves equations that have no neat algebraic solution by repeating a process: you put an estimate in, get a better estimate out, and feed it back in. The values usually converge towards a root.
Rearranging to an iterative formula
An equation is rearranged into the form xₙ₊₁ = f(xₙ), making x the subject in a way that can be repeated. The same equation can give different iterative formulae, some converging faster than others.
Applying the formula step by step
Start with x₀, substitute it to get x₁, then substitute x₁ to get x₂, and so on. Each value is the input for the next. Keep full calculator accuracy between steps to avoid errors building up.
Using the ANS button
The ANS button makes iteration quick: type the formula using ANS in place of xₙ, enter your starting value, then press equals repeatedly. Each press gives the next iteration without retyping.
Convergence and accuracy
As you iterate, the values get closer together, converging on the root. You stop when consecutive values agree to the required accuracy (e.g. the same to 3 decimal places). The iteration is approximating where a graph crosses the x-axis.
Worked examples
Example 1: First iteration
Using xₙ₊₁ = √(xₙ + 2) with x₀ = 2, find x₁.
x₁ = √(2 + 2) = √4 = 2.
Example 2: Second iteration
Using xₙ₊₁ = (xₙ + 5)/2 with x₀ = 3, find x₂.
x₁ = (3 + 5)/2 = 4; x₂ = (4 + 5)/2 = 4.5.
Example 3: When to stop
Consecutive values are 1.521 and 1.521 to 3 d.p. What does this mean?
They agree to 3 d.p., so the root is 1.521 (3 d.p.).
Common mistakes and how to avoid them
Rounding between steps. Keep full accuracy until the end.
Forgetting to feed back. Each output becomes the next input.
Stopping too early. Continue until values agree to the required accuracy.
Mis-typing the formula. Use brackets and ANS carefully.
Wrong starting value. Use the one given (or a sensible estimate near the root).
Exam technique for Iteration
Substitute x₀ to begin, then feed each result back in.
Use the ANS button for speed and accuracy.
Keep full accuracy between iterations.
Stop when values agree to the required number of decimal places.
Relate the root to where the graph crosses the axis.
Quick revision summary
Iteration finds approximate solutions to equations that cannot be solved exactly by repeating a process: an estimate goes in and a better one comes out, usually converging on a root. The equation is rearranged into the form xₙ₊₁ = f(xₙ), and you start with x₀, substitute to get x₁, then feed each value back in to get the next. The ANS button speeds this up — type the formula with ANS, enter the start value, and press equals repeatedly. Keep full calculator accuracy between steps, and stop when consecutive values agree to the required accuracy (e.g. the same to 3 d.p.). The iteration is approximating where the graph crosses the x-axis. The common errors are rounding between steps, forgetting to feed the result back, stopping too early, and mis-typing the formula. Substitute the start value, feed results back, use ANS, keep full accuracy, and stop when the values settle.