GCSE Maths Iteration Questions With Worked Solutions
Iteration means using the result of one calculation as the input for the next. In GCSE Higher Maths, questions often give a recurrence formula and ask you to calculate successive approximations to a root. The arithmetic is usually straightforward; the challenge is following the formula precisely, keeping enough decimal places and knowing what the values mean.
Understand the notation for iteration
A formula such as x(n+1) = √(x(n) + 1) says that the next value is the square root of the current value plus 1. The subscript n labels the iteration number, not a power. If the starting value is x0 = 1, then x1 = √(1 + 1) = √2, approximately 1.41421356. The second value uses that whole new result: x2 = √(1.41421356 + 1), approximately 1.55377397. Continue this process rather than returning to the starting value at every step.
Work through a GCSE Maths iteration table
Using x(n+1) = √(x(n) + 1) and x0 = 1, the first few values are x1 ≈ 1.414, x2 ≈ 1.554, x3 ≈ 1.598, x4 ≈ 1.612 and x5 ≈ 1.616. The outputs approach roughly 1.618. A table with columns n and x(n) helps maintain the correct order. If your calculator allows an answer-memory function, use it carefully so the unrounded previous result feeds the next step. When writing the table, show an appropriate number of digits to make the approach clear.
Why the iteration approaches a root
When the values settle to a limit L, the recurrence suggests L = √(L + 1). Squaring gives L² = L + 1, so L² − L − 1 = 0. The positive solution is (1 + √5)/2, about 1.618. This explains the equation being solved by the iteration. However, simply rearranging an equation does not guarantee every recurrence will converge from every starting value. In GCSE questions, use the specified iteration formula and initial value rather than inventing an untested rearrangement.
Use enough precision at every stage
Rounding x1 to 1.4 before substituting it into the formula for x2 introduces an avoidable error that can accumulate. Keep the calculator’s full internal value during the calculation sequence and round only when recording the values requested. If the question asks for answers to three decimal places, display the values correctly but use unrounded numbers for the next iteration wherever possible. A sequence may appear stable to two decimal places while still changing in the fourth or fifth. The required accuracy determines when you can reasonably stop.
Recognise common iteration mistakes
Do not confuse x(n+1) with x(n) + 1 outside the square root; the brackets determine what is calculated first. Avoid pressing the square key when the expression requires square root. A recurring error is copying x2 where x3 should be, or applying the formula twice before entering the next row. If a recurrence uses division, check the denominator and bracket placement. Estimate whether the next value should rise or fall; an unexpectedly huge result is often a calculator-entry problem.
Check your final root estimate
Substitute the approximate value into the original equation, not only the rearranged recurrence. For L ≈ 1.618, L² − L − 1 is close to zero, supporting the estimate. State the result to the requested degree of accuracy and avoid claiming an exact answer from a rounded decimal. If your values are still changing at the requested precision, continue iterating. A strong answer shows the starting value, the recurrence, the required intermediate results and an explicit statement of the approximate root.
Read also: GCSE Maths Grade 3: Which Skills Come First?
Frequently asked questions
Why are repeated values allowed to approach a solution?
A convergent recurrence can generate successive approximations that become arbitrarily close to a fixed point.
Can every equation be solved by any iteration formula?
No. Convergence depends on the chosen rearrangement and starting value. Follow the formula specified in the exam question.
More GCSE learning resources
Explore these existing guides when a prerequisite skill or another aspect of your GCSE preparation needs attention. Use the links selectively to find the explanation that matches your current problem.
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