- Writing the final correct expression A1"}]},{"id":"0b488ad1-ad93-44f0-bd5e-cd7c2a428cd8:simultaneous_equations","label":"Solving Simultaneous Equations","steps":[{"type":"text","order":0,"title":"Set up general form","content":"Any quadratic sequence can be written in the general form:\n\n$$u_n = an^2 + bn + c$$\n\nwhere $a$, $b$, and $c$ are constants we need to find, and $u_n$ represents the $n$th term of the sequence."},{"type":"hyper_latex","block":{"nodes":[{"text":"Using the first three terms of our sequence, we can set up a system of equations by substituting $n = 1$, $n = 2$, and $n = 3$ into the general formula:"},{"latex":"$$\\begin{aligned} \\text{For } n = 1: \\quad a(1)^2 + b(1) + c &= 6 \\implies \\htmlData{anchor=eq1}{\\color{@sky}{a + b + c = 6}} \\\\ \\text{For } n = 2: \\quad a(2)^2 + b(2) + c &= 23 \\implies \\htmlData{anchor=eq2}{\\color{@amber}{4a + 2b + c = 23}} \\\\ \\text{For } n = 3: \\quad a(3)^2 + b(3) + c &= 48 \\implies \\htmlData{anchor=eq3}{\\color{@purple}{9a + 3b + c = 48}} \\end{aligned}$"}],"highlights":[{"color":"sky","style":"box","anchor":"eq1"},{"color":"amber","style":"box","anchor":"eq2"},{"color":"purple","style":"box","anchor":"eq3"}],"connections":[]},"order":1,"title":"Substitute term values","description":"This gives us a system of three linear equations with three variables: $a$, $b$, and $c$."},{"type":"text","order":2,"title":"Subtract the equations","content":"To eliminate the constant $c$, we can subtract our equations from one another:\n\nSubtract Equation (1) from Equation (2):\n\n$$(4a + 2b + c) - (a + b + c) = 23 - 6$$\n$$3a + b = 17 \\quad \\text{--- Equation (4)}$$\n\nSubtract Equation (2) from Equation (3):\n\n$$(9a + 3b + c) - (4a + 2b + c) = 48 - 23$$\n$$5a + b = 25 \\quad \\text{--- Equation (5)}$$"},{"type":"text","order":3,"title":"Solve for a","content":"Now we have a system of two equations with two variables ($a$ and $b$). Subtract Equation (4) from Equation (5) to eliminate $b$:\n\n$$(5a + b) - (3a + b) = 25 - 17$$\n$$2a = 8$$\n$$a = 4$$"},{"type":"text","order":4,"title":"Solve for b and c","content":"Substitute $a = 4$ back into Equation (4) to find $b$:\n\n$$3(4) + b = 17$$\n$$12 + b = 17$$\n$$b = 5$$\n\nNow, substitute $a = 4$ and $b = 5$ into Equation (1) to find $c$:\n\n$$4 + 5 + c = 6$$\n$$9 + c = 6$$\n$$c = -3$$"},{"type":"text","order":5,"title":"Write final expression","content":"Substitute the values of $a = 4$, $b = 5$, and $c = -3$ back into our general form $an^2 + bn + c$:\n\n$$\\text{Nth term} = 4n^2 + 5n - 3$$"}]}],"generatedAt":"2026-06-12T19:47:17.212Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"987d3892-c7ae-477a-940e-023a9bd07c07","revisionId":"609d55b3-fc24-41ee-98ec-869dc2148b6e","questionNumber":null,"questionType":"LA","specification":"Here are the first 5 terms of a quadratic sequence.\n\n16 24 34 46 60","difficulty":4,"difficultyBand":"hard","totalMarks":6,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"62b75899-2d14-41bf-a797-bfdc52ecc378","label":"a","content":"Show that the $n$th term is $n^2 + 5n + 10$","markscheme":"- First differences: 8, 10, 12, 14\n- Second difference: 2\n- $2a = 2 \\implies a = 1$ M1\n- $3a + b = 8 \\implies 3(1) + b = 8 \\implies b = 5$ M1\n- $a + b + c = 16 \\implies 1 + 5 + c = 16 \\implies c = 10$ M1\n- Conclusion: $n^2 + 5n + 10$ as required A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Calculates the second difference and finds the coefficient of $n^2$ ($a=1$).","code":"M"},{"id":"p2","marks":1,"description":"Uses a correct method to find the coefficient of $n$ ($b=5$).","code":"M"},{"id":"p3","marks":1,"description":"Uses a correct method to find the constant term ($c=10$).","code":"M"},{"id":"p4","marks":1,"description":"Correctly states the full $n$th term expression.","code":"A"}],"finalAnswers":["$$n^2 + 5n + 10$"],"generatedAt":"2026-05-06T14:02:37.167Z","modelVersion":"gemini-3-flash-preview"

The Nth Term of a Quadratic Sequence

Question 9

The Nth Term of a Quadratic Sequence Questions

The Nth Term of a Quadratic Sequence

Question 9

The Nth Term of a Quadratic Sequence Questions