Home   >  Worksheets   >  GCSE questions by topic   >  Iteration

MathsLinks Logo Questions by topic

Last update: 05/11/2026

Give feedback : Buy me a coffee

Iteration

Free Printable Iteration PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 7)

Download free iteration worksheets (PDF), GCSE maths past paper questions and worked solutions for Iteration (7K). This is a Grade 7 algebra topic from the GCSE questions by topic collection for Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Corbett Maths, Maths Made Easy, Naiker Maths, 1st Class Maths, Metatutor, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our iteration revision notes and worked examples with diagrams, try real GCSE past paper iteration questions with step-by-step solutions, or download a PDF of all iteration worksheet links.

Free Iteration PDF Worksheets & Answers

Grade 7 • Topic 7K
Resource Questions Answers
GCSE past paper questions & solutions
 Qs Ans
Graded exam questions & solutions
 Qs Ans
Practice Questions & answers
 Qs Ans
MME worksheet & model answers
 Qs Ans
Exam question booklet & solutions
 Qs Ans
Edexcel exam questions & solutions
 Qs Ans
AQA exam questions & solutions
 Qs Ans
Topic worksheet & worked solutions
 Qs Ans
Practice questions & answers
 Qs Ans
Practice questions & answers
 Qs Ans
Practice questions & answers
 Qs Ans
GCSE topic booklet & answers
 Qs Ans
PMT (Edxl)
Edexcel GCSE topic questions & solutions
 Qs Ans
PMT (AQA)
AQA GCSE topic questions & solutions
 Qs Ans
Interactive online lesson & practice
 Page
Self-marking online worksheet
 Page
Grade 7 topics
7A - Surds7B - Bounds
7C - Proportion (harder)7D - Algebraic fractions
7E - The quadratic formula7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs7H - Rearranging (harder)
7J - Functions (harder)7K - Iteration
7L - Area of any triangle7M - The sine rule
7N - The cosine rule7P - Congruent triangles
7Q - 3D pythagoras / trig.7R - Histograms
7S - Conditional probability7T - Exact trig. values (harder)
7U - Area of a segment
GCSE Maths Grade 7 Revision Notes & Method

Guide to Iteration: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

To show that an equation f(x) = 0 has a root between x = a and x = b, evaluate f(a) and f(b), state that there is a change of sign (one positive, one negative) and f(x) is continuous. An iterative formula xn+1 = g(xn) is formed by rearranging f(x) = 0 to isolate one x term.

To iterate efficiently on a calculator, type the starting value x0 and press =, then enter the formula using the ANS key in place of xn. Press = repeatedly to generate x1, x2, x3, writing down full calculator displays at each step.

Example 1: Show x3 + 2x − 6 = 0 has a root between 1 and 2, and rearrange to x = ∛(6 − 2x).

Solution:

Let f(x) = x3 + 2x − 6:
f(1) = 13 + 2(1) − 6 = −3 (< 0)
f(2) = 23 + 2(2) − 6 = +6 (> 0)
Change of sign shows a root lies in (1, 2).
Rearrange: x3 = 6 − 2x ⇒ x = ∛(6 − 2x)

Example 2: Use xn+1 = ∛(6 − 2xn) with x0 = 1.5 to find x1, x2, x3 (3 d.p.).

Solution:

Type 1.5 =, then ∛(6 − 2 × ANS):
x1 = ∛(6 − 2(1.5)) = ∛3 = 1.4422...
x2 = ∛(6 − 2(1.4422...)) = 1.4605...
x3 = ∛(6 − 2(1.4605...)) = 1.4548...
Answer: x1 = 1.442, x2 = 1.461, x3 = 1.455
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Iteration Questions & Answers

Past Paper Example 1 GCSE Jun 2023 2H Q17 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
(a) Show that x⁴ – x² – 5 = 0 can be written as x = ⁴√(x² + 5). (1)
(b) Starting with x₀ = 1.5, use the iteration formula three times to find x₃. (3)
(a) Show that the equation x4 - x2 - 5 = 0 can be written in the form x = 4√(x2 + 5) [1 mark]
(b) Starting with x0 = 1.5
use the iteration formula xn+1 = 4√(xn2 + 5) three times to find an estimate for a solution of x4 - x2 - 5 = 0 [3 marks]
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
x4 = x2 + 5 ⇒ x = 4√(x2 + 5)
Answer (a): x4 = x2 + 5 ⇒ x = 4√(x2 + 5)
Part (b) Solution:
x₁ = ⁴√(1.5² + 5) = 1.640909... M1
x₂ = ⁴√(x₁² + 5) = 1.665418... M1
x₃ = ⁴√(x₂² + 5) = 1.669763... A1
Answer (b): 1.66976
Past Paper Example 2 GCSE Jun 2017 3H Q16 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
Using xn+1 = -2 - 4xn2
with x0 = -2.5
(a) find the values of x1, x2 and x3 [3 marks]
(b) Explain the relationship between the values of x1, x2 and x3 and the equation x3 + 2x2 + 4 = 0 [2 marks]
Worked Solution & Mark Scheme (5 marks)
Part (a) Solution:
x₁ = –2 – 4/(–2.5)² = –2.64 M1
x₂ = –2 – 4/(–2.64)² = –2.57392... M1
x₃ = –2 – 4/(–2.57392...)² = –2.603767 A1
Answer (a): x₁ = -2.64, x₂ = -2.57392, x₃ = -2.60377
Part (b) Solution:
x = –2 – 4/x² ⇔ x³ + 2x² + 4 = 0, so the iterates converge to the root M1 C1
Answer (b): The iterative formula is a rearrangement of x³ + 2x² + 4 = 0 (dividing by x² gives x + 2 + 4/x² = 0 ⇒ x = –2 – 4/x²), so x₁, x₂ and x₃ are successive estimates of the solution to the equation.
Past Paper Example 3 AQA Jun 2024 2H Q22 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
@@TABULAR0@@
Use x1 = 1 to work out an approximate solution to x = 5 - 1x
Give your answer to 4 significant figures.
Worked Solution & Mark Scheme (3 marks)
x₂ = 5 – 1/1 = 4. M1
x₃ = 5 – 1/4 = 4.75.
x₄ = 5 – 1/4.75 = 4.78947... M1
x₅ = 5 – 1/4.78947... = 4.79120...
x₆ = 4.79128... → 4.791 (to 4 s.f.). A1
Answer: x = 4.791

← 7J Functions (harder)

Back to GCSE questions by topic

7L Area of any triangle →