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# Iteration
- URL: https://www.esheets.io/iteration/
- Published: 2026-10-17T18:56:35.000Z
- Updated: 2026-10-17T19:09:10.000Z
- Author: Richard Linnington
- Tags: Maths, Algebra

Iteration uses a repeated calculation to produce a sequence of values. A suitable convergent iteration can produce increasingly accurate estimates of a solution. This worksheet develops calculator fluency with recurrence rules, including negative starting values, fractions and powers, before using iteration to estimate solutions of equations.

You will also practise rearranging an equation into an iteration rule and using a sign change to show that a root lies within an interval. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Eight calculator questions build from generating terms of a recurrence to estimating roots of nonlinear equations.

Each question is worth one mark. Where a question has several answer boxes, all of them must be correct for the mark.

#### What you’ll practise

- using subscripts and repeated substitution accurately;
- working with negative values, powers and rational expressions;
- rearranging equations into iteration form and interpreting convergence;
- showing that a root is bracketed by a sign change, then estimating it.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked examples.

Use a calculator and the previous full-precision value at each step. Give answers to 3 decimal places only where stated.

## Topic guide

### What iteration means

An iterative rule repeats the same calculation. The starting value is **x0**, **xn** means the current value, and **xn+1** means the next value. To find x1, substitute x0; to find x2, substitute the full calculator value of x1; then continue in the same way.

### A numerical iteration

For xn+1 \= 0.4xn \+ 3 and x0 \= 5:

- x1 \= 0.4 × 5 + 3 = 5
- x2 \= 0.4 × 5 + 3 = 5

This sequence has reached a fixed value. Other sequences approach a limiting value gradually. That behaviour is called **convergence**.

### Calculator accuracy

Keep every intermediate value in the calculator. Round only the requested displayed answers. Rounding after each step can change the final estimate. With negative inputs, use brackets before squaring or cubing: for example, (−2)2 \= 4 but −22 is interpreted as −4 by many calculators.

### Rearranging to make an iteration rule

To solve x3 \+ 20x = 30, one possible rearrangement is

x = 30 − x320.

This suggests xn+1 \= (30 − xn3)/20\. Starting with x0 \= 1.4 gives x1 \= 1.3628 and x2 ≈ 1.3734, using the unrounded x1. The values are moving towards a solution of the original equation. A rearrangement that is algebraically correct is not automatically useful: the chosen rule and starting value must produce a convergent sequence.

### Showing that a root lies in an interval

For a continuous function f(x), calculate the values at both ends of the interval. If f(a) and f(b) have opposite signs, the graph crosses the x-axis somewhere between a and b, so there is at least one root in that interval.

For example, let f(x) = x3 \+ x − 7\. Then f(1) = −5 and f(2) = 3\. The sign change shows that a root lies between 1 and 2\. A suitable rule is xn+1 \= ∛(7 − xn). Starting from a stated value in the interval and repeating the rule gives an estimate of that root.

### Common mistakes

- Using x0 again for every line instead of using the previous result.
- Giving x2 when the question asks for x3.
- Rounding intermediate calculator values.
- Forgetting brackets around a negative value before applying a power.
- Claiming a sign change when both function values have the same sign.