Iteration

Gold-standard guide

26 mins

What you will learn

Use repeated calculations to approximate solutions.

Use a clear step-by-step method for iteration.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier algebra skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Iteration: substitute the current approximation into the right-hand side to get the next one

Step 1

Calculate x_1 from x_0 = 2

x_1 = $\sqrt{3 + 2}$ = $\sqrt{5}$ ≈ 2.2360679…

Step 2

Calculate x_2 from x_1

x_2 = $\sqrt{3 + 2.2360679}$ = $\sqrt{5.2360679}$ ≈ 2.2882456…

Step 3

Calculate x_3 from x_2

x_3 = $\sqrt{3 + 2.2882456}$ = $\sqrt{5.2882456}$ ≈ 2.2996521…

Watch out

Students round at every stage, compounding rounding errors

f

Iteration

Use the previous answer as the next input.

f

Accuracy

Round only at the end unless the question tells you otherwise.

Worked example

Using the iterative formula x_{n+1} = $\sqrt{3 + x_n}$ with x_0 = 2, find x_3 to 3 decimal places.

1

Calculate x_1 from x_0 = 2: x_1 = $\sqrt{3 + 2}$ = $\sqrt{5}$ ≈ 2.2360679…

2

Calculate x_2 from x_1: x_2 = $\sqrt{3 + 2.2360679}$ = $\sqrt{5.2360679}$ ≈ 2.2882456…

3

Calculate x_3 from x_2: x_3 = $\sqrt{3 + 2.2882456}$ = $\sqrt{5.2882456}$ ≈ 2.2996521…

4

Round only the final answer: x_3 ≈ 2.300 to 3 d.p.

Final answer

x_3 ≈ 2.300

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Using the iterative formula x_{n+1} = $\sqrt{3 + x_n}$ with x_0 = 2, find x_3 to 3 decimal places.

4 marks4 minsiteration-worked

Show solution

Worked solution

1.Calculate x_1 from x_0 = 2: x_1 = $\sqrt{3 + 2}$ = $\sqrt{5}$ ≈ 2.2360679…
2.Calculate x_2 from x_1: x_2 = $\sqrt{3 + 2.2360679}$ = $\sqrt{5.2360679}$ ≈ 2.2882456…
3.Calculate x_3 from x_2: x_3 = $\sqrt{3 + 2.2882456}$ = $\sqrt{5.2882456}$ ≈ 2.2996521…
4.Round only the final answer: x_3 ≈ 2.300 to 3 d.p.

Final answer

x_3 ≈ 2.300

Mark points

M1: calculate x_1 from x_0 = 2
M1: calculate x_2 from x_1
M1: calculate x_3 from x_2
M1: round only the final answer
A1: x_3 ≈ 2.300

Watch out

Students round at every stage, compounding rounding errors.Store the full calculator display and only round the final answer.In the exam, write the unrounded value alongside the rounded one so the examiner can follow your working.

Diagnosticrecall

Use x_{n+1} = (x_n² + 5)/6 with x_0 = 1 to find x_2 to 3 d.p.

1 mark2 minsiteration-q1

Show solution

Worked solution

1.Spot the skill: Iteration: substitute the current approximation into the right-hand side to get the next one.
2.Use the calculate x_1 from x_0 = 2 stage first, then calculate x_2 from x_1.
3.Keep the final answer visible: x_1 = 1, x_2 = 1.000 (converges immediately — fixed point).

Final answer

x_1 = 1, x_2 = 1.000 (converges immediately — fixed point)

Mark points

M1: use the correct iteration: substitute the current approximation into the right-hand side to get the next one.repeat until consecutive values agree to the required degree of accuracy.the solution being found is where the curve y = f(x) crosses y = x.
A1: x_1 = 1, x_2 = 1.000 (converges immediately — fixed point)

Watch out

Students round at every stage, compounding rounding errors.Store the full calculator display and only round the final answer.In the exam, write the unrounded value alongside the rounded one so the examiner can follow your working.

Easyprocedure

Use x_{n+1} = $\sqrt{x_n + 6}$ with x_0 = 3. Find x_2 to 3 d.p.

2 marks3 minsiteration-q2

Show solution

Worked solution

1.Spot the skill: Iteration: substitute the current approximation into the right-hand side to get the next one.
2.Use the calculate x_2 from x_1 stage first, then calculate x_3 from x_2.
3.Keep the final answer visible: x_1 = 3, x_2 = 3.000.

Final answer

x_1 = 3, x_2 = 3.000

Mark points

M1: use the correct iteration: substitute the current approximation into the right-hand side to get the next one.repeat until consecutive values agree to the required degree of accuracy.the solution being found is where the curve y = f(x) crosses y = x.
A1: x_1 = 3, x_2 = 3.000

Watch out

Students round at every stage, compounding rounding errors.Store the full calculator display and only round the final answer.In the exam, write the unrounded value alongside the rounded one so the examiner can follow your working.

Mediumreasoning

Show that x³ + x = 10 has a root between x = 1 and x = 2.

3 marks4 minsiteration-q3

Show solution

Worked solution

1.Spot the skill: Iteration: substitute the current approximation into the right-hand side to get the next one.
2.Use the calculate x_3 from x_2 stage first, then round only the final answer.
3.Keep the final answer visible: f(1) = 2 > 0 and... wait — f(1) = 1+1−10 = −8 < 0 and f(2) = 8+2−10 = 0; root at x = 2.

Final answer

f(1) = 2 > 0 and... wait — f(1) = 1+1−10 = −8 < 0 and f(2) = 8+2−10 = 0; root at x = 2

Mark points

M1: use the correct iteration: substitute the current approximation into the right-hand side to get the next one.repeat until consecutive values agree to the required degree of accuracy.the solution being found is where the curve y = f(x) crosses y = x.
A1: f(1) = 2 > 0 and... wait — f(1) = 1+1−10 = −8 < 0 and f(2) = 8+2−10 = 0; root at x = 2

Watch out

Students round at every stage, compounding rounding errors.Store the full calculator display and only round the final answer.In the exam, write the unrounded value alongside the rounded one so the examiner can follow your working.

Hardproblem solving

Use x_{n+1} = 10/(x_n² + 1), x_0 = 2. Find x_2.

3 marks5 minsiteration-q4

Show solution

Worked solution

1.Spot the skill: Iteration: substitute the current approximation into the right-hand side to get the next one.
2.Use the round only the final answer stage first, then calculate x_1 from x_0 = 2.
3.Keep the final answer visible: x_1 = 2, x_2 = 2 (if converges); actually x_1 = $1\frac{0}{5}$ = 2.000.

Final answer

x_1 = 2, x_2 = 2 (if converges); actually x_1 = $1\frac{0}{5}$ = 2.000

Mark points

M1: use the correct iteration: substitute the current approximation into the right-hand side to get the next one.repeat until consecutive values agree to the required degree of accuracy.the solution being found is where the curve y = f(x) crosses y = x.
A1: x_1 = 2, x_2 = 2 (if converges); actually x_1 = $1\frac{0}{5}$ = 2.000

Watch out

Students round at every stage, compounding rounding errors.Store the full calculator display and only round the final answer.In the exam, write the unrounded value alongside the rounded one so the examiner can follow your working.

Exam-stylemulti-step

An iterative formula gives x_4 = 3.4721 and x_5 = 3.4719. State the root to 3 d.p.

4 marks6 minsiteration-q5

Show solution

Worked solution

1.Spot the skill: Iteration: substitute the current approximation into the right-hand side to get the next one.
2.Use the calculate x_1 from x_0 = 2 stage first, then calculate x_2 from x_1.
3.Keep the final answer visible: 3.472.

Final answer

3.472

Mark points

M1: use the correct iteration: substitute the current approximation into the right-hand side to get the next one.repeat until consecutive values agree to the required degree of accuracy.the solution being found is where the curve y = f(x) crosses y = x.
A1: 3.472

Watch out

Students round at every stage, compounding rounding errors.Store the full calculator display and only round the final answer.In the exam, write the unrounded value alongside the rounded one so the examiner can follow your working.

Grade 9 stretchproblem solving

Using x_(n+1) = $\sqrt{7 + x_n}$ and x_0 = 2, find x_2 to 3 decimal places.

4 marks7 minsiteration-g9

Show solution

Worked solution

1.Calculate x_1.
2.Use that answer in the rule to calculate x_2.
3.Round only at the end.

Final answer

x_2 = 3.162

Mark points

M1: x_1 = 3
M1: x_2 = $\sqrt{10}$
A1: x_2 = 3.162

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Iteration - 2 marksUse x_{n+1} = (x_n² + 5)/6 with x_0 = 1 to find x_2 to 3 d.p.Mark answer

Answer

x_1 = 1, x_2 = 1.000 (converges immediately — fixed point)

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer

Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer

Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer

Answer

3x(2x + 3)

Iteration repeatedly feeds the answer back in

What you will learn

Know the rule, then use it

Method

Calculate x_1 from x_0 = 2

Calculate x_2 from x_1

Calculate x_3 from x_2

Watch out

Using the iterative formula x_{n+1} = $\sqrt{3 + x_n}$ with x_0 = 2, find x_3 to 3 decimal places.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

Iteration

Iteration repeatedly feeds the answer back in

What you will learn

Know the rule, then use it

Method

Calculate x_1 from x_0 = 2

Calculate x_2 from x_1

Calculate x_3 from x_2

Watch out

Using the iterative formula x_{n+1} = 3+xn\sqrt{3 + x_n}3+xn​​ with x_0 = 2, find x_3 to 3 decimal places.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

Using the iterative formula x_{n+1} = $\sqrt{3 + x_n}$ with x_0 = 2, find x_3 to 3 decimal places.