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1\\end{aligned}$$","difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"b1ae92ed-f5ad-46ae-9d82-884e3191ef35","label":"a","content":"Find an expression for $x_2$ in terms of $a$.","markscheme":"Substitutes $x_1 = 2$ into the recurrence relation: $x_2 = a(2) - 5$ M1\nSimplifies to $2a - 5$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Substitutes $x_1 = 2$ into the recurrence relation to obtain $x_2 = a(2) - 5$.","code":"M","acceptable":["$$x_2 = a(2) - 5$","$$x_2 = 2a - 5$"]},{"id":"p2","marks":1,"description":"Simplifies to $x_2 = 2a - 5$.","code":"A","acceptable":["$$2a - 5$","$$x_2 = 2a - 5$"]}],"finalAnswers":["$$2a - 5$"],"generatedAt":"2026-09-22T10:32:14.262Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"e8c13297-1727-46c5-aa58-604b2bb037d1","label":"b","content":"Show that $x_3 = 2a^2 - 5a - 5$.","markscheme":"Substitutes their expression for $x_2$ into the recurrence relation: $x_3 = a(2a - 5) - 5$ M1\nExpands and simplifies to reach the given answer: $2a^2 - 5a - 5$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Substitutes $x_2 = 2a - 5$ into the recurrence relation to obtain $x_3 = a(2a - 5) - 5$.","code":"M","acceptable":["$$x_3 = a(2a - 5) - 5$","$$x_3 = a x_2 - 5$ with $x_2 = 2a - 5$"]},{"id":"p2","marks":1,"description":"Expands and simplifies to show that $x_3 = 2a^2 - 5a - 5$.","code":"A","acceptable":["$$2a^2 - 5a - 5$","$$x_3 = 2a^2 - 5a - 5$"]}],"finalAnswers":["$$2a^2 - 5a - 5$"],"generatedAt":"2026-09-22T10:32:14.262Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"f720a587-5196-4994-bc04-8c4b71229630","label":"c","content":"Given that $a = -\\frac{5}{2}$ or $a = 5$, find the possible values of $x_3$.","markscheme":"Substitutes $a = -\\frac{5}{2}$ and $a = 5$ into $x_3 = 2a^2 - 5a - 5$ M1\nCalculates $x_3 = 2\\left(-\\frac{5}{2}\\right)^2 - 5\\left(-\\frac{5}{2}\\right) - 5 = 20$ for $a = -\\frac{5}{2}$ M1\nCalculates $x_3 = 2(5)^2 - 5(5) - 5 = 20$ for $a = 5$, giving the possible value $x_3 = 20$ A1","marks":3,"order":2,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Substitutes $a = -\\frac{5}{2}$ and $a = 5$ into $x_3 = 2a^2 - 5a - 5$.","code":"M","acceptable":["$$x_3 = 2\\left(-\\frac{5}{2}\\right)^2 - 5\\left(-\\frac{5}{2}\\right) - 5$ and $x_3 = 2(5)^2 - 5(5) - 5$","Substitution of both given values into the expression for $x_3$"]},{"id":"p2","marks":1,"description":"Correctly calculates $x_3 = 20$ when $a = -\\frac{5}{2}$.","code":"M","acceptable":["$$2\\left(-\\frac{5}{2}\\right)^2 - 5\\left(-\\frac{5}{2}\\right) - 5 = 20$","$$x_3 = 20$ for $a = -\\frac{5}{2}$"]},{"id":"p3","marks":1,"description":"Correctly calculates $x_3 = 20$ when $a = 5$ and gives the possible value $x_3 = 20$.","code":"A","acceptable":["$$2(5)^2 - 5(5) - 5 = 20$","$$x_3 = 20$"]}],"finalAnswers":["$$x_3 = 20$"],"generatedAt":"2026-09-22T10:32:14.262Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"b22a61de-33f0-4195-890b-da105e0c7cd2","revisionId":"538d141c-fbd0-4969-aa49-fa6edc85ac86","questionNumber":null,"questionType":"LA","specification":"The first term of a geometric series is 5 and the common ratio is 0.75.","difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"6775f565-0784-4998-94d1-d32e2c5f1b43","label":"a","content":"Find the second term and the sum to infinity.","markscheme":"$$5(0.75)=3.75$ B1\n\n$\\frac{5}{1-0.75}=20$ B1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds the second term using $ar$","code":"B","acceptable":["$$3.75$","$$5(0.75)=3.75$"]},{"id":"p2","marks":1,"description":"Finds the sum to infinity using $\\frac{a}{1-r}$","code":"B","acceptable":["$$20$","$$\\frac{5}{1-0.75}=20$"]}],"finalAnswers":["3.75 and 20"],"generatedAt":"2026-09-22T10:53:51.283Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"6ff06ad2-7b34-4bc2-b095-30c2f9946a2c","label":"b","content":"Find the smallest value of $n$ for which $S_n > 19$.","markscheme":"$$S_n=\\frac{5(1-0.75^n)}{1-0.75}=20(1-0.75^n)$ B1\n\n$20(1-0.75^n)>19$ M1\n\n$0.75^n<0.05\\implies n>\\log_{0.75}0.05\\approx10.4$ M1\n\n$n=11$ A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Uses the sum of the first $n$ terms correctly","code":"B","acceptable":["$$S_n=\\frac{5(1-0.75^n)}{1-0.75}$","$$S_n=20(1-0.75^n)$"]},{"id":"p2","marks":1,"description":"Forms the correct inequality","code":"M","acceptable":["$$20(1-0.75^n)>19$"]},{"id":"p3","marks":1,"description":"Solves the inequality for $n$","code":"M","acceptable":["$$0.75^n<0.05$","$$n>\\log_{0.75}0.05\\approx10.4$"]},{"id":"p4","marks":1,"description":"Chooses the smallest integer value of $n$","code":"A","acceptable":["$$n=11$"]}],"finalAnswers":["11"],"generatedAt":"2026-09-22T10:53:51.283Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"ba1ac045-e145-4b7a-b4f3-2592764a560b","revisionId":"d7e6ce49-c1be-4ba8-add6-311a0dac262f","questionNumber":null,"questionType":"LA","specification":"A sequence $x_1, x_2, x_3, \\dots$ is defined by $x_1 = 2$ and $x_{n+1} = ax_n - 5, \\quad n \\geq 1$.","difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"da30b785-d57b-4002-8eea-191fb39e74c5","label":"a","content":"Find an expression for $x_2$ in terms of $a$.","markscheme":"$$x_2 = a(2) - 5 = 2a - 5$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Finds $x_2$ in terms of $a$.","code":"B","acceptable":["$$x_2 = a(2) - 5 = 2a - 5$","$$2a - 5$"]}],"finalAnswers":["$$2a - 5$"],"generatedAt":"2026-09-22T10:41:25.596Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"93ced7ea-be91-49f2-8450-b45bd226cd3e","label":"b","content":"Show that $x_3 = 2a^2 - 5a - 5$.","markscheme":"$$x_3 = a(2a-5) - 5$ M1\n$x_3 = 2a^2 - 5a - 5$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Uses the recurrence to obtain $x_3 = a(2a-5)-5$.","code":"M","acceptable":["$$x_3 = a(2a-5)-5$"]},{"id":"p2","marks":1,"description":"Expands and simplifies to the required expression.","code":"A","acceptable":["$$x_3 = 2a^2 - 5a - 5$"]}],"finalAnswers":["$$2a^2 - 5a - 5$"],"generatedAt":"2026-09-22T10:41:25.596Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"225c9669-e387-4825-bbc2-420059eded64","label":"c","content":"Given that $a = -2.5$ or $a = 5$, find the possible values of $x_3$.","markscheme":"For $a=-2.5$ or $a=5$, $x_3 = 2a^2 - 5a - 5$ M1\n$a=-2.5: \\; x_3=2(-2.5)^2-5(-2.5)-5=20$ and $a=5: \\; x_3=2(5)^2-5(5)-5=20$ M1\n$x_3=20$ A1","marks":3,"order":2,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Uses $x_3 = 2a^2 - 5a - 5$ for the given values of $a$.","code":"M","acceptable":["Substitutes $a=-2.5$ and $a=5$ into $x_3 = 2a^2 - 5a - 5$"]},{"id":"p2","marks":1,"description":"Correctly evaluates $x_3$ for both given values of $a$.","code":"M","acceptable":["For $a=-2.5$, obtains $x_3=20$, 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Sequences and Series

Question 119

Sequences and Series Questions

Sequences and Series

Question 119

Sequences and Series Questions