Lesson A20

A20 Solving equations by iteration Quiz: AQA Maths, Unit 2

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson A20, Solving equations by iteration: 20 multiple choice questions for the AQA GCSE Maths (8300), Unit 2: Algebra, written with Revision Ninja.

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The 20 questions

  1. (H) Starting at x0 = 2, x(n+1) = x(n)^2 - 3, what is x1?

    • -1
    • 1
    • 4
    • 7

  2. (H) What does x(n+1) denote in an iterative method?

    • The value of x(n) squared
    • The gradient of the curve at x(n)
    • The exact root of the equation
    • The next approximation, found from x(n)

  3. (H) Using x(n+1) = 2 + 1/x(n) with x0 = 2, what is x1?

    • 2.5
    • 3
    • 1.5
    • 2.4

  4. (H) Using x(n+1) = 2 + 1/x(n) with x0 = 2, what is x2 to 1 decimal place?

    • 2.5
    • 2.25
    • 3.4
    • 2.4

  5. Between which two whole numbers does the root of x^3 + x - 4 = 0 lie?

    • 2 and 3
    • -1 and 0
    • 1 and 2
    • 0 and 1

  6. A sign change of f(x) between x = 1 and x = 2 shows what?

    • There are no roots in this interval
    • The graph has a turning point at x = 1
    • The root is exactly 1.5
    • A root lies between 1 and 2

  7. (H) For x(n+1) = sqrt(x(n) + 1) with x0 = 1, what is x1 to 2 decimal places?

    • 1.5
    • 1.00
    • 2.00
    • 1.41

  8. (H) When successive iterates settle down to the same value, what does this mean?

    • They satisfy the rearranged equation
    • The graph has a maximum there
    • The equation has no real roots
    • The method has failed

  9. In x(n+1), what does the subscript n + 1 indicate?

    • The next term after the nth term in the sequence of iterates
    • The gradient of the graph at the point x = n + 1
    • The term n plus one in a different sequence of values
    • The first term multiplied by n, repeated each step

  10. (H) Starting with x0 = 3, what is x1 for x(n+1) = 5 - x(n)?

    • 2
    • 3
    • 8
    • -3

  11. (H) For x(n+1) = 4 / x(n) with x0 = 1, what is x2?

    • 0.25
    • 2
    • 4
    • 1

  12. (H) To iterate x^3 + 2x = 10, which rearrangement is suitable?

    • x = (10 - x^3) / 2
    • x = (10 - 2x) / x^3
    • x = 10 - x^3 + 2
    • x = 10 / (x^3 + 2)

  13. (H) Why might iterating a rearrangement fail to find a root?

    • The starting value must always be zero
    • Each iterate must be a whole number
    • Iteration only works for linear equations
    • Its iterates may move away from the root

  14. (H) Starting at x0 = 1, x(n+1) = 1 + 1/x(n), what is x2?

    • 1.67
    • 1.5
    • 2
    • 1

  15. (H) Iterating x(n+1) = 0.5 x(n) + 3 from x0 = 0, what is x2?

    • 3
    • 4.5
    • 6
    • 1.5

  16. (H) The iteration x(n+1) = 3 / x(n) from x0 = 1 gives 1, 3, 1, 3 and so on. Why does it not converge?

    • The iterates alternate and never settle on a value
    • The iterates reach zero after two steps and stop
    • The iterates grow without limit, getting larger each step
    • The iterates equal the root of x^2 + 3

  17. (H) f(x) = x^2 - 5 has f(2) < 0 and f(3) > 0. Between which values is the root?

    • 2 and 3
    • 3 and 4
    • -3 and -2
    • 0 and 2

  18. (H) To find a root to 2 decimal places by iteration, when should you stop?

    • When the function value is exactly 2 at that step
    • When two successive iterates agree to 2 decimal places
    • After five iterations, regardless of the values reached
    • When the iterate equals zero to two decimal places

  19. (H) Starting at x0 = 3, x(n+1) = 2 x(n) - 1, what is x2?

    • 9
    • 5
    • 11
    • 7

  20. (H) Starting at x0 = 4, x(n+1) = x(n) / 2 + 1, what is x3?

    • 3
    • 2
    • 2.5
    • 2.25

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