ck":{"nodes":[{"latex":"$$x_3=2+\\dfrac{3}{\\color{@sky}{\\htmlData{anchor=previous}{x_2}}}$"},{"latex":"$$x_3=2+\\dfrac{3}{\\color{@sky}{\\htmlData{anchor=substituted}{2.6}}}$"},{"latex":"$$\\color{@green}{x_3=3.153846154}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"previous"},{"color":"sky","style":"text-color","anchor":"substituted"}],"connections":[{"to":"substituted","from":"previous","color":"sky","label":"substitute"}]},"order":3,"title":"Calculate the third term","description":"Use the exact second term in the denominator. Giving the decimal shown earns the final B1 mark."}]}],"generatedAt":"2026-07-19T07:17:57.786Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"91cc88cb-f92d-4524-b092-9b97bccbc1fe","label":"b","content":"Explain the relationship between the values of $x_1$, $x_2$ and $x_3$ and the equation $x^2 - 2x - 3 = 0$","markscheme":"- Shows rearrangement: $x = 2 + \\frac{3}{x} \\Rightarrow x - 2 = \\frac{3}{x} \\Rightarrow x(x - 2) = 3 \\Rightarrow x^2 - 2x - 3 = 0$ C1\n- States that $x_1, x_2, x_3$ are successive approximations or estimates of a root/solution to the equation. C1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correct algebraic rearrangement of the iterative formula into the target quadratic equation.","code":"C"},{"id":"p2","marks":1,"description":"Identification that the values of $x_n$ are estimates or approximations of a solution to the equation.","code":"C"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:02:02.470Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"fixed_point_iteration","label":"Link the iteration to the equation","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=fixed}{\\color{@sky}{x=2+\\dfrac{3}{x}}}$"},{"latex":"$$\\htmlData{anchor=subtract}{\\color{@orange}{x-2=\\dfrac{3}{x}}}$"},{"latex":"$$\\htmlData{anchor=multiply}{\\color{@green}{x(x-2)=3}}$"},{"latex":"$$\\htmlData{anchor=quadratic}{\\color{@purple}{x^2-2x-3=0}}$"}],"highlights":[{"color":"purple","style":"text-color-with-background","anchor":"quadratic"}],"connections":[{"to":"subtract","from":"fixed","color":"orange","label":"subtract 2"},{"to":"multiply","from":"subtract","color":"green","label":"multiply by x"},{"to":"quadratic","from":"multiply","color":"purple","label":"expand and rearrange"}]},"order":1,"title":"Rearrange the fixed point","description":"When the iteration settles, consecutive values are treated as the same value, x. Completing this rearrangement to the quadratic earns the first C1 mark."},{"type":"text","order":2,"title":"State the relationship","content":"Therefore, $x_1$, $x_2$ and $x_3$ are **successive approximations** (increasingly updated estimates) of a root of $x^2-2x-3=0$. Here they are approximating the root $x=3$. This statement earns the second C1 mark; do not say that all three values are roots, because they are only estimates.","highlights":[{"text":"Explain the relationship between the values of $x_1$, $x_2$ and $x_3$ and the equation $x^2 - 2x - 3 = 0$","location":"specification"}]}]}],"generatedAt":"2026-07-19T07:36:56.665Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"d36e0a7f-dbfd-4616-8c2a-843742a85c2e","revisionId":"de3a48e5-80c1-4538-8e53-a69ce5b8b927","questionNumber":null,"questionType":"LA","specification":"Using $x_{n+1} = 5 + \\frac{4}{x_n^2}$ with $x_0 = 2$","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"calc","parts":[{"id":"1d29b758-26b0-4d61-97aa-557488540ef3","label":"a","content":"Find the values of $x_1$, $x_2$ and $x_3$.","markscheme":"- $x_1 = 5 + \\frac{4}{(2)^2} = 6$ B1\n- $x_2 = 5 + \\frac{4}{(6)^2} = 5.111111111$ B1\n- $x_3 = 5 + \\frac{4}{(5.111111111)^2} = 5.153119093$ B1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correct

Iteration

Question 7

Iteration Questions

Iteration

Question 7

Iteration Questions