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only)\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/ccea/sequences-and-series-4/questions/q145\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/ccea/questions","basePath":"/a-level/maths/ccea/sequences-and-series-4/questions","formulaUrl":"https://ccea.org.uk/downloads/docs/Support/General/2019/Mathematical%20Formulae%20and%20Tables.pdf","initialIndex":144,"initialQuestionWindow":[{"id":"98d41dcb-08fe-42f2-aa9f-92f939ea317e","revisionId":"3db702e1-48e7-4a28-b785-d1fe747a4f9d","questionNumber":null,"questionType":"LA","specification":"A subscription-based fitness app tracks the number of new premium users each month. In the first three months after a major update, the number of new premium users, in thousands, was recorded as:\n\n\\[ 3k - 28, \\quad k, \\quad k + 6 \\]\n\nwhere \$k\$ is a constant. Given that the number of new users joining each month is modeled as a geometric series,","difficulty":5,"difficultyBand":"medium","tier":null,"totalMarks":9,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"7de09d2e-751b-428a-90d7-0caec45f59dd","label":"a","content":"show that \$k^2 - 5k - 84 = 0\$.","markscheme":"- Since the terms are in a geometric progression, the common ratio $r$ is constant: $\\dfrac{k}{3k - 28} = \\dfrac{k + 6}{k}$ M1\n- Cross-multiply to obtain $k^2 = (3k - 28)(k + 6) = 3k^2 + 18k - 28k - 168$ M1\n- Rearrange to $2k^2 - 10k - 168 = 0$ and divide by 2 to reach $k^2 - 5k - 84 = 0$ A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"common_ratio","label":"Equating Common Ratios","steps":[{"type":"text","order":0,"title":"Equate the common ratios","content":"For three consecutive terms in a geometric sequence, the ratio between successive terms is constant. This is called the common ratio, $r$.\n\nWe can find two expressions for $r$ using our three terms, $3k - 28$, $k$, and $k + 6$:\n\n1. Divide the second term by the first term:\n$$r = \\frac{k}{3k - 28}$$\n\n2. Divide the third term by the second term:\n$$r = \\frac{k + 6}{k}$$\n\nSince the common ratio is the same, we can equate these two expressions:\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Cross-multiply and expand","content":"To eliminate the fractions, we cross-multiply both sides of the equation:\n\n$$k \\times k = (3k - 28)(k + 6)$$\n\n$$k^2 = (3k - 28)(k + 6)$$\n\nNow, expand the right-hand side using the FOIL method:\n\n$$k^2 = 3k^2 + 18k - 28k - 168$$\n\nSimplify the terms on the right-hand side:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":2,"title":"Rearrange and simplify","content":"Rearrange the equation to set it equal to zero by subtracting $k^2$ from both sides:\n\n$$0 = 2k^2 - 10k - 168$$\n\nTo match the target equation, we can divide every term in the equation by $2$:\n\n$$0 = k^2 - 5k - 84$$\n\nRearranging gives us the desired quadratic equation:\n\n$$k^2 - 5k - 84 = 0$$"}]},{"id":"geometric_mean","label":"Using Geometric Mean Relation","steps":[{"type":"text","order":0,"title":"Identify terms of the sequence","content":"We are given that the first three terms of a geometric series are:\n\n* $u_1 = 3k - 28$\n* $u_2 = k$\n* $u_3 = k + 6$\n\nwhere $k$ is a constant."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_1 = \\color{@sky}{3k - 28}, \\quad u_2 = \\color{@purple}{k}, \\quad u_3 = \\color{@amber}{k + 6}$"},{"latex":"$$(\\htmlData{anchor=u2-lhs}{\\color{@purple}{u_2}})^2 = \\htmlData{anchor=u1-rhs}{\\color{@sky}{u_1}} \\cdot \\htmlData{anchor=u3-rhs}{\\color{@amber}{u_3}}$"},{"latex":"$$(\\htmlData{anchor=k-lhs}{\\color{@purple}{k}})^2 = (\\htmlData{anchor=u1-sub}{\\color{@sky}{3k - 28}})(\\htmlData{anchor=u3-sub}{\\color{@amber}{k + 6}})$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"u2-lhs"},{"color":"sky","style":"text-color","anchor":"u1-rhs"},{"color":"amber","style":"text-color","anchor":"u3-rhs"}],"connections":[{"to":"k-lhs","from":"u2-lhs","color":"purple","label":"substitute","style":"solid"},{"to":"u1-sub","from":"u1-rhs","color":"sky","label":null,"style":"solid"},{"to":"u3-sub","from":"u3-rhs","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Apply geometric mean relation","description":"In any geometric progression, the ratio between consecutive terms is constant, which means the square of the middle term is equal to the product of the two outer terms: $u_2^2 = u_1 u_3$. This is called the geometric mean relation."},{"type":"text","order":2,"title":"Expand and simplify the equation","content":"Now, let's expand the right-hand side of the equation:\n\n$$(3k - 28)(k + 6) = 3k^2 + 18k - 28k - 168$$\n\nCombine the like terms in the expansion:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":3,"title":"Rearrange to the final form","content":"To get our quadratic equation into standard form, we move all terms to one side:\n\n$$3k^2 - k^2 - 10k - 168 = 0$$\n\n$$2k^2 - 10k - 168 = 0$$\n\nNow, divide every term by $2$ to simplify to the final requested equation:\n\n$$k^2 - 5k - 84 = 0$$\n\nThis completes the proof as required."}]}],"generatedAt":"2026-06-09T08:04:47.713Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"763732e7-bab0-4f62-a4cf-28baaedd9d5e","label":"b","content":"Predict the number of new premium users joining in the 6th month.","markscheme":"- Solve $k^2 - 5k - 84 = 0 \\Rightarrow (k - 12)(k + 7) = 0$. Since user counts must be positive, $k = 12$ M1\n- Find $a = 3(12) - 28 = 8$ and $r = \\frac{12}{8} = 1.5$. Then $u_6 = 8 \\times (1.5)^{5} = 60.75$. The number of users is $60,750$ A1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"algebraic_formula","label":"Algebraic Equation and Formula","steps":[{"type":"text","order":0,"title":"Set up the ratio equation","content":"For a sequence to be geometric, the ratio between consecutive terms must be constant. This constant is the common ratio, $r$.\n\nUsing the first three terms of our sequence, we can write:\n\n$$r = \\frac{u_2}{u_1} = \\frac{u_3}{u_2}$$\n\nSubstituting the terms given in the question, $u_1 = 3k - 28$, $u_2 = k$, and $u_3 = k + 6$, we get:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"To solve this equation, cross-multiply to eliminate the fractions:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpand the right-hand side of the equation:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$\n\nRearrange everything to one side to form a standard quadratic equation:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ simplifies it to:\n\n$$k^2 - 5k - 84 = 0$$"},{"type":"text","order":2,"title":"Solve for $k$","content":"Now we factorise the quadratic equation to find the values of $k$. We are looking for two numbers that multiply to $-84$ and add up to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible solutions for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of new premium users must be positive, the term $u_2 = k$ must be positive. Therefore, we discard $k = -7$ and select:\n\n$$k = 12$$"},{"type":"text","order":3,"title":"Find the first term and ratio","content":"Now substitute $k = 12$ back to find the first term, $a$ (or $u_1$), and the common ratio, $r$:\n\n* First term: \n $$a = 3(12) - 28 = 36 - 28 = 8$$\n* Common ratio: \n $$r = \\frac{k}{a} = \\frac{12}{8} = 1.5$$"},{"type":"text","order":4,"title":"Calculate the 6th term","content":"The general formula for the $n$-th term of a geometric sequence is:\n\n$$u_n = a r^{n-1}$$\n\nTo find the number of users in the 6th month ($u_6$):\n\n$$u_6 = 8 \\times (1.5)^{6-1}$$\n\n$$u_6 = 8 \\times (1.5)^5$$\n\n$$u_6 = 8 \\times 7.59375 = 60.75$$"},{"type":"text","order":5,"title":"Convert to the final unit","content":"The question specifies that the number of premium users is tracked **in thousands**. \n\nTherefore, we multiply our result by $1,000$ to get the final prediction:\n\n$$\\text{New premium users} = 60.75 \\times 1,000 = 60,750$$"}]},{"id":"recursive_calculation","label":"Step-by-Step Multiplication","steps":[{"type":"text","order":0,"title":"Set up the equation for $k$","content":"Since the number of new users joining each month forms a geometric sequence, the ratio between consecutive terms must be constant.\n\nThis means the common ratio $r$ can be written as:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{\\text{Term 3}}{\\text{Term 2}}$$\n\nSubstituting our terms $3k - 28$, $k$, and $k + 6$:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$\n\nNow, we cross-multiply to solve for $k$:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpanding the right-hand side:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":1,"title":"Solve the quadratic equation","content":"Rearranging the equation to standard quadratic form $ax^2 + bx + c = 0$ by subtracting $k^2$ from both sides:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ to simplify:\n\n$$k^2 - 5k - 84 = 0$$\n\nNow we factorise this quadratic. We need two numbers that multiply to $-84$ and add to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible values for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of users joining must be positive (for example, Term 2 is $k$), we must choose the positive value:\n\n$$k = 12$$","calculator":[{"gdc":{"instructions":"Go to the Equation menu, select Polynomial (F2), choose Degree 2, enter coefficients 1, -5, and -84, then press SOLVE (F6)."},"ti84":{"instructions":"Press [APPS], select 'PlySmlt2' (Polynomial Root Finder and Simultaneous Equation Solver). Select 'Poly Root Finder', set order to 2, and enter coefficients: a = 1, b = -5, c = -84. Press [SOLVE]."},"desmos":{"expression":"y = x^2 - 5x - 84","instructions":"Graph the equation y = x^2 - 5x - 84 and identify the x-intercepts."},"description":"Verify the roots of the quadratic equation $k^2 - 5k - 84 = 0$ using a solver."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$k = \\htmlData{anchor=k-val}{\\color{@sky}{12}}$"},{"latex":"$$\\text{Term 1: } 3(\\htmlData{anchor=k-sub1}{\\color{@sky}{12}}) - 28 = \\htmlData{anchor=t1}{\\color{@amber}{8}}$"},{"latex":"$$\\text{Term 2: } \\htmlData{anchor=t2}{\\color{@purple}{12}}$"},{"latex":"$$\\text{Term 3: } \\htmlData{anchor=k-sub3}{\\color{@sky}{12}} + 6 = \\htmlData{anchor=t3}{\\color{@green}{18}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"k-val"},{"color":"amber","style":"box","anchor":"t1"},{"color":"purple","style":"box","anchor":"t2"},{"color":"green","style":"box","anchor":"t3"}],"connections":[{"to":"k-sub1","from":"k-val","color":"sky","label":"Substitute","style":"solid"},{"to":"k-sub3","from":"k-val","color":"sky","label":"Substitute","style":"solid"}]},"order":2,"title":"Find the first three terms","description":"Now, we substitute $k = 12$ back into our expressions for the first three months to find the actual user numbers (in thousands)."},{"type":"text","order":3,"title":"Find the common ratio","content":"To find the common ratio $r$, we divide any term by its preceding term:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{12}{8} = 1.5$$\n\nLet's double check this with the next terms:\n\n$$r = \\frac{\\text{Term 3}}{\\text{Term 2}} = \\frac{18}{12} = 1.5$$\n\nSo, the common ratio is $r = 1.5$. Each month, the number of new premium users increases by a factor of $1.5$."},{"type":"text","order":4,"title":"Multiply sequentially to Month 6","content":"To find the number of new premium users joining in the 6th month, we can multiply step-by-step starting from Month 3:\n\n- **Month 3:** $18$ thousand\n- **Month 4:** $18 \\times 1.5 = 27$ thousand\n- **Month 5:** $27 \\times 1.5 = 40.5$ thousand\n- **Month 6:** $40.5 \\times 1.5 = 60.75$ thousand"},{"type":"text","order":5,"title":"State the final answer","content":"Since the terms are given in thousands, we convert our Month 6 value back into the actual number of users:\n\n$$60.75 \\times 1,000 = 60,750$$\n\nTherefore, the predicted number of new premium users joining in the 6th month is **$60,750$**."}]}],"generatedAt":"2026-06-09T08:05:25.252Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"b3bb30d1-85aa-41ba-83b9-c94256e3a632","label":"c","content":"The total number of all premium users who have joined the app since the update is predicted to exceed 5 million for the first time in the \$N\$th month.\n\nCalculate the value of \$N\$ according to the model.","markscheme":"- Identify that the sum $S_N$ must exceed 5000 (since 5 million is 5000 thousand) B1\n- Use $S_N = \\dfrac{a(r^N - 1)}{r - 1}$ to set up the inequality: $\\dfrac{8(1.5^N - 1)}{1.5 - 1} > 5000$ M1\n- Simplify to $16(1.5^N - 1) > 5000 \\Rightarrow 1.5^N > 313.5$ and apply logarithms: $N \\ln 1.5 > \\ln 313.5$ M1\n- Solve $N > \\dfrac{\\ln 313.5}{\\ln 1.5} \\approx 14.17$, so $N = 15$ A1","marks":4,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"analytical_logarithms","label":"Analytical Approach using Logarithms","steps":[{"type":"text","order":0,"title":"Convert units and identify parameters","content":"First, we must align the units of our values. The question states that the number of new premium users is modeled in **thousands**.\n\nTherefore, we convert the target of 5 million premium users into thousands:\n\n$$\\text{Target} = 5,000,000 = 5000 \\text{ thousand}$$\n\nFrom the previous parts of this problem, we have the model parameters for the geometric series:\n- First term, $a = 8$\n- Common ratio, $r = 1.5$\n\nWe need to find the first month $N$ such that the sum of the first $N$ terms, $S_N$, exceeds $5000$ thousand:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum inequality","content":"Recall the formula for the sum of the first $N$ terms of a geometric series from the formula booklet:\n\n$$S_N = \\frac{a(r^N - 1)}{r - 1}$$\n\nSubstitute our parameters $a = 8$ and $r = 1.5$ into the sum formula and set it to be greater than $5000$:\n\n$$\\frac{8(1.5^N - 1)}{1.5 - 1} > 5000$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom}{\\color{@sky}{1.5 - 1}}} > 5000$"},{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom-val}{\\color{@sky}{0.5}}} > 5000$"},{"latex":"$$\\htmlData{anchor=coeff}{\\color{@amber}{16}}(1.5^N - 1) > 5000$"},{"latex":"$$1.5^N - 1 > \\frac{5000}{\\color{@amber}{16}}$"},{"latex":"$$1.5^N - 1 > \\htmlData{anchor=frac-val}{\\color{@purple}{312.5}}$"},{"latex":"$$1.5^N > \\htmlData{anchor=final-val}{\\color{@teal}{313.5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"denom"},{"color":"amber","style":"text-color","anchor":"coeff"},{"color":"purple","style":"text-color","anchor":"frac-val"},{"color":"teal","style":"text-color","anchor":"final-val"}],"connections":[{"to":"denom-val","from":"denom","color":"sky","label":"simplify","style":"solid"},{"to":"coeff","from":"denom-val","color":"amber","label":"8 / 0.5 = 16","style":"solid"},{"to":"frac-val","from":"coeff","color":"purple","label":"divide","style":"solid"},{"to":"final-val","from":"frac-val","color":"teal","label":"+ 1","style":"solid"}]},"order":2,"title":"Simplify the inequality","description":"We simplify the inequality step-by-step by dividing out the common factor and adding 1 to both sides."},{"type":"text","order":3,"title":"Apply logarithms to solve for N","content":"To solve for $N$ analytically when it is in the exponent, we take the natural logarithm ($\n\\ln$) of both sides of the inequality:\n\n$$\\ln(1.5^N) > \\ln(313.5)$$\n\nUsing the power rule of logarithms, $\\ln(x^y) = y \\ln(x)$, we can bring the exponent $N$ to the front:\n\n$$N \\ln(1.5) > \\ln(313.5)$$\n\nSince $1.5 > 1$, $\\ln(1.5)$ is a positive number, so we can divide both sides by $\\ln(1.5)$ without reversing the inequality sign:\n\n$$N > \\frac{\\ln(313.5)}{\\ln(1.5)}$$"},{"type":"text","order":4,"title":"Calculate the final month N","content":"Now we evaluate the expression using a calculator:\n\n$$N > 14.172\\dots$$\n\nSince the month $N$ must be a whole number, we find the first integer value of $N$ that satisfies this inequality. \n\nThe first integer greater than $14.17$ is **$15$**.\n\nThus, the total number of premium users is predicted to exceed 5 million for the first time in the **$15$th** month.","calculator":[{"gdc":{"expression":"ln(313.5)/ln(1.5)","instructions":"Enter the logarithmic division formula directly into the run-matrix menu to find the decimal limit."},"ti84":{"expression":"ln(313.5)/ln(1.5)","instructions":"Press [LN], type 313.5, press [)], press [/], press [LN], type 1.5, press [)], then press [ENTER]."},"desmos":{"expression":"\\frac{\\ln(313.5)}{\\ln(1.5)}","instructions":"Type the expression in a blank cell to find the value of N."},"description":"Calculate the exact analytical limit for $N$ using logarithms."}]}]},{"id":"gdc_table","label":"Calculator Table or Graphical Solver","steps":[{"type":"text","order":0,"title":"Convert 5 million to thousands","content":"Since the number of new premium users is given in **thousands**, we first need to convert 5 million into thousands so that the units of our equation match. \n\n$$\\text{Target} = \\frac{5,000,000}{1,000} = 5000 \\text{ thousand}$$ \n\nWe want to find the smallest integer month $N$ such that the sum of new users up to month $N$ is strictly greater than $5000$:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum formula","content":"We recall the formula for the sum of the first $n$ terms of a geometric series from the formula booklet:\n\n$$S_n = \\frac{a(r^n - 1)}{r - 1}$$\n\nGiven the first term $a = 8$ and the common ratio $r = 1.5$, we substitute these values into the formula to express $S_N$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{1.5 - 1}$$\n\nSimplifying the denominator $1.5 - 1 = 0.5$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{0.5} = 16(1.5^N - 1)$$\n\nSo, our inequality is:\n\n$$16(1.5^N - 1) > 5000$$"},{"type":"text","order":2,"title":"Analyze using calculator table","content":"To find the first integer value of $N$ where $S_N > 5000$, we can use our graphing calculator's table feature to compute $S_N$ for successive values of $N$. Let's define the function:\n\n$$Y_1 = 16(1.5^x - 1)$$\n\nNow, generate the table of values to observe when $Y_1$ first exceeds $5000$.","calculator":[{"gdc":{"expression":"16*(1.5^x - 1)","instructions":"1. Go to the Graph/Table menu.\n2. Enter the function \\\$f1(x) = 16(1.5^x - 1)\\\$.\n3. Open the table of values (ctrl + T on TI-Nspire).\n4. Scroll through the x-values to find when the function value exceeds 5000."},"ti84":{"expression":"16*(1.5^X - 1)","instructions":"1. Press [Y=] and enter \\\$Y_1 = 16(1.5^X - 1)\\\$.\n2. Press [2nd] [WINDOW] (TBLSET). Set TblStart = 10 and \\\$\\Delta\\\$Tbl = 1.\n3. Press [2nd] [GRAPH] (TABLE) to view the table of values.\n4. Scroll down to find the first row where \\\$Y_1 > 5000\\\$."},"desmos":{"expression":"f(x) = 16*(1.5^x - 1)","instructions":"1. Type \\\$f(x) = 16(1.5^x - 1)\\\$ in the input bar.\n2. Click the gear icon (Tools) and select the \"Convert to table\" icon.\n3. Input values of x from 12 to 16 to inspect the corresponding outputs."},"description":"Generate a table of values for the sum function to find when the sum exceeds 5000."}]},{"type":"text","order":3,"title":"Evaluate table values","content":"Using the table of values, we find:\n\n- For $N = 13$:\n $$S_{13} = 16(1.5^{13} - 1) \\approx 16(194.62 - 1) \\approx 3097.9$$\n- For $N = 14$:\n $$S_{14} = 16(1.5^{14} - 1) \\approx 16(291.93 - 1) \\approx 4654.9$$\n- For $N = 15$:\n $$S_{15} = 16(1.5^{15} - 1) \\approx 16(437.89 - 1) \\approx 6990.3$$\n\nThus, the total predicted premium users exceed 5 million ($5000$ thousand) for the first time during month $15$."},{"type":"text","order":4,"title":"State final answer","content":"The first month in which the total number of premium users exceeds 5 million is **15**.\n\n$$N = 15$$"}]}],"generatedAt":"2026-06-09T08:06:11.884Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"7de09d2e-751b-428a-90d7-0caec45f59dd:common_ratio","label":"(a) Equating Common Ratios","steps":[{"type":"text","order":0,"title":"Equate the common ratios","content":"For three consecutive terms in a geometric sequence, the ratio between successive terms is constant. This is called the common ratio, $r$.\n\nWe can find two expressions for $r$ using our three terms, $3k - 28$, $k$, and $k + 6$:\n\n1. Divide the second term by the first term:\n$$r = \\frac{k}{3k - 28}$$\n\n2. Divide the third term by the second term:\n$$r = \\frac{k + 6}{k}$$\n\nSince the common ratio is the same, we can equate these two expressions:\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Cross-multiply and expand","content":"To eliminate the fractions, we cross-multiply both sides of the equation:\n\n$$k \\times k = (3k - 28)(k + 6)$$\n\n$$k^2 = (3k - 28)(k + 6)$$\n\nNow, expand the right-hand side using the FOIL method:\n\n$$k^2 = 3k^2 + 18k - 28k - 168$$\n\nSimplify the terms on the right-hand side:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":2,"title":"Rearrange and simplify","content":"Rearrange the equation to set it equal to zero by subtracting $k^2$ from both sides:\n\n$$0 = 2k^2 - 10k - 168$$\n\nTo match the target equation, we can divide every term in the equation by $2$:\n\n$$0 = k^2 - 5k - 84$$\n\nRearranging gives us the desired quadratic equation:\n\n$$k^2 - 5k - 84 = 0$$"}]},{"id":"7de09d2e-751b-428a-90d7-0caec45f59dd:geometric_mean","label":"(a) Using Geometric Mean Relation","steps":[{"type":"text","order":0,"title":"Identify terms of the sequence","content":"We are given that the first three terms of a geometric series are:\n\n* $u_1 = 3k - 28$\n* $u_2 = k$\n* $u_3 = k + 6$\n\nwhere $k$ is a constant."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_1 = \\color{@sky}{3k - 28}, \\quad u_2 = \\color{@purple}{k}, \\quad u_3 = \\color{@amber}{k + 6}$"},{"latex":"$$(\\htmlData{anchor=u2-lhs}{\\color{@purple}{u_2}})^2 = \\htmlData{anchor=u1-rhs}{\\color{@sky}{u_1}} \\cdot \\htmlData{anchor=u3-rhs}{\\color{@amber}{u_3}}$"},{"latex":"$$(\\htmlData{anchor=k-lhs}{\\color{@purple}{k}})^2 = (\\htmlData{anchor=u1-sub}{\\color{@sky}{3k - 28}})(\\htmlData{anchor=u3-sub}{\\color{@amber}{k + 6}})$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"u2-lhs"},{"color":"sky","style":"text-color","anchor":"u1-rhs"},{"color":"amber","style":"text-color","anchor":"u3-rhs"}],"connections":[{"to":"k-lhs","from":"u2-lhs","color":"purple","label":"substitute","style":"solid"},{"to":"u1-sub","from":"u1-rhs","color":"sky","label":null,"style":"solid"},{"to":"u3-sub","from":"u3-rhs","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Apply geometric mean relation","description":"In any geometric progression, the ratio between consecutive terms is constant, which means the square of the middle term is equal to the product of the two outer terms: $u_2^2 = u_1 u_3$. This is called the geometric mean relation."},{"type":"text","order":2,"title":"Expand and simplify the equation","content":"Now, let's expand the right-hand side of the equation:\n\n$$(3k - 28)(k + 6) = 3k^2 + 18k - 28k - 168$$\n\nCombine the like terms in the expansion:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":3,"title":"Rearrange to the final form","content":"To get our quadratic equation into standard form, we move all terms to one side:\n\n$$3k^2 - k^2 - 10k - 168 = 0$$\n\n$$2k^2 - 10k - 168 = 0$$\n\nNow, divide every term by $2$ to simplify to the final requested equation:\n\n$$k^2 - 5k - 84 = 0$$\n\nThis completes the proof as required."}]},{"id":"763732e7-bab0-4f62-a4cf-28baaedd9d5e:algebraic_formula","label":"(b) Algebraic Equation and Formula","steps":[{"type":"text","order":0,"title":"Set up the ratio equation","content":"For a sequence to be geometric, the ratio between consecutive terms must be constant. This constant is the common ratio, $r$.\n\nUsing the first three terms of our sequence, we can write:\n\n$$r = \\frac{u_2}{u_1} = \\frac{u_3}{u_2}$$\n\nSubstituting the terms given in the question, $u_1 = 3k - 28$, $u_2 = k$, and $u_3 = k + 6$, we get:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"To solve this equation, cross-multiply to eliminate the fractions:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpand the right-hand side of the equation:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$\n\nRearrange everything to one side to form a standard quadratic equation:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ simplifies it to:\n\n$$k^2 - 5k - 84 = 0$$"},{"type":"text","order":2,"title":"Solve for $k$","content":"Now we factorise the quadratic equation to find the values of $k$. We are looking for two numbers that multiply to $-84$ and add up to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible solutions for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of new premium users must be positive, the term $u_2 = k$ must be positive. Therefore, we discard $k = -7$ and select:\n\n$$k = 12$$"},{"type":"text","order":3,"title":"Find the first term and ratio","content":"Now substitute $k = 12$ back to find the first term, $a$ (or $u_1$), and the common ratio, $r$:\n\n* First term: \n $$a = 3(12) - 28 = 36 - 28 = 8$$\n* Common ratio: \n $$r = \\frac{k}{a} = \\frac{12}{8} = 1.5$$"},{"type":"text","order":4,"title":"Calculate the 6th term","content":"The general formula for the $n$-th term of a geometric sequence is:\n\n$$u_n = a r^{n-1}$$\n\nTo find the number of users in the 6th month ($u_6$):\n\n$$u_6 = 8 \\times (1.5)^{6-1}$$\n\n$$u_6 = 8 \\times (1.5)^5$$\n\n$$u_6 = 8 \\times 7.59375 = 60.75$$"},{"type":"text","order":5,"title":"Convert to the final unit","content":"The question specifies that the number of premium users is tracked **in thousands**. \n\nTherefore, we multiply our result by $1,000$ to get the final prediction:\n\n$$\\text{New premium users} = 60.75 \\times 1,000 = 60,750$$"}]},{"id":"763732e7-bab0-4f62-a4cf-28baaedd9d5e:recursive_calculation","label":"(b) Step-by-Step Multiplication","steps":[{"type":"text","order":0,"title":"Set up the equation for $k$","content":"Since the number of new users joining each month forms a geometric sequence, the ratio between consecutive terms must be constant.\n\nThis means the common ratio $r$ can be written as:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{\\text{Term 3}}{\\text{Term 2}}$$\n\nSubstituting our terms $3k - 28$, $k$, and $k + 6$:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$\n\nNow, we cross-multiply to solve for $k$:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpanding the right-hand side:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":1,"title":"Solve the quadratic equation","content":"Rearranging the equation to standard quadratic form $ax^2 + bx + c = 0$ by subtracting $k^2$ from both sides:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ to simplify:\n\n$$k^2 - 5k - 84 = 0$$\n\nNow we factorise this quadratic. We need two numbers that multiply to $-84$ and add to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible values for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of users joining must be positive (for example, Term 2 is $k$), we must choose the positive value:\n\n$$k = 12$$","calculator":[{"gdc":{"instructions":"Go to the Equation menu, select Polynomial (F2), choose Degree 2, enter coefficients 1, -5, and -84, then press SOLVE (F6)."},"ti84":{"instructions":"Press [APPS], select 'PlySmlt2' (Polynomial Root Finder and Simultaneous Equation Solver). Select 'Poly Root Finder', set order to 2, and enter coefficients: a = 1, b = -5, c = -84. Press [SOLVE]."},"desmos":{"expression":"y = x^2 - 5x - 84","instructions":"Graph the equation y = x^2 - 5x - 84 and identify the x-intercepts."},"description":"Verify the roots of the quadratic equation $k^2 - 5k - 84 = 0$ using a solver."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$k = \\htmlData{anchor=k-val}{\\color{@sky}{12}}$"},{"latex":"$$\\text{Term 1: } 3(\\htmlData{anchor=k-sub1}{\\color{@sky}{12}}) - 28 = \\htmlData{anchor=t1}{\\color{@amber}{8}}$"},{"latex":"$$\\text{Term 2: } \\htmlData{anchor=t2}{\\color{@purple}{12}}$"},{"latex":"$$\\text{Term 3: } \\htmlData{anchor=k-sub3}{\\color{@sky}{12}} + 6 = \\htmlData{anchor=t3}{\\color{@green}{18}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"k-val"},{"color":"amber","style":"box","anchor":"t1"},{"color":"purple","style":"box","anchor":"t2"},{"color":"green","style":"box","anchor":"t3"}],"connections":[{"to":"k-sub1","from":"k-val","color":"sky","label":"Substitute","style":"solid"},{"to":"k-sub3","from":"k-val","color":"sky","label":"Substitute","style":"solid"}]},"order":2,"title":"Find the first three terms","description":"Now, we substitute $k = 12$ back into our expressions for the first three months to find the actual user numbers (in thousands)."},{"type":"text","order":3,"title":"Find the common ratio","content":"To find the common ratio $r$, we divide any term by its preceding term:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{12}{8} = 1.5$$\n\nLet's double check this with the next terms:\n\n$$r = \\frac{\\text{Term 3}}{\\text{Term 2}} = \\frac{18}{12} = 1.5$$\n\nSo, the common ratio is $r = 1.5$. Each month, the number of new premium users increases by a factor of $1.5$."},{"type":"text","order":4,"title":"Multiply sequentially to Month 6","content":"To find the number of new premium users joining in the 6th month, we can multiply step-by-step starting from Month 3:\n\n- **Month 3:** $18$ thousand\n- **Month 4:** $18 \\times 1.5 = 27$ thousand\n- **Month 5:** $27 \\times 1.5 = 40.5$ thousand\n- **Month 6:** $40.5 \\times 1.5 = 60.75$ thousand"},{"type":"text","order":5,"title":"State the final answer","content":"Since the terms are given in thousands, we convert our Month 6 value back into the actual number of users:\n\n$$60.75 \\times 1,000 = 60,750$$\n\nTherefore, the predicted number of new premium users joining in the 6th month is **$60,750$**."}]},{"id":"b3bb30d1-85aa-41ba-83b9-c94256e3a632:analytical_logarithms","label":"(c) Analytical Approach using Logarithms","steps":[{"type":"text","order":0,"title":"Convert units and identify parameters","content":"First, we must align the units of our values. The question states that the number of new premium users is modeled in **thousands**.\n\nTherefore, we convert the target of 5 million premium users into thousands:\n\n$$\\text{Target} = 5,000,000 = 5000 \\text{ thousand}$$\n\nFrom the previous parts of this problem, we have the model parameters for the geometric series:\n- First term, $a = 8$\n- Common ratio, $r = 1.5$\n\nWe need to find the first month $N$ such that the sum of the first $N$ terms, $S_N$, exceeds $5000$ thousand:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum inequality","content":"Recall the formula for the sum of the first $N$ terms of a geometric series from the formula booklet:\n\n$$S_N = \\frac{a(r^N - 1)}{r - 1}$$\n\nSubstitute our parameters $a = 8$ and $r = 1.5$ into the sum formula and set it to be greater than $5000$:\n\n$$\\frac{8(1.5^N - 1)}{1.5 - 1} > 5000$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom}{\\color{@sky}{1.5 - 1}}} > 5000$"},{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom-val}{\\color{@sky}{0.5}}} > 5000$"},{"latex":"$$\\htmlData{anchor=coeff}{\\color{@amber}{16}}(1.5^N - 1) > 5000$"},{"latex":"$$1.5^N - 1 > \\frac{5000}{\\color{@amber}{16}}$"},{"latex":"$$1.5^N - 1 > \\htmlData{anchor=frac-val}{\\color{@purple}{312.5}}$"},{"latex":"$$1.5^N > \\htmlData{anchor=final-val}{\\color{@teal}{313.5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"denom"},{"color":"amber","style":"text-color","anchor":"coeff"},{"color":"purple","style":"text-color","anchor":"frac-val"},{"color":"teal","style":"text-color","anchor":"final-val"}],"connections":[{"to":"denom-val","from":"denom","color":"sky","label":"simplify","style":"solid"},{"to":"coeff","from":"denom-val","color":"amber","label":"8 / 0.5 = 16","style":"solid"},{"to":"frac-val","from":"coeff","color":"purple","label":"divide","style":"solid"},{"to":"final-val","from":"frac-val","color":"teal","label":"+ 1","style":"solid"}]},"order":2,"title":"Simplify the inequality","description":"We simplify the inequality step-by-step by dividing out the common factor and adding 1 to both sides."},{"type":"text","order":3,"title":"Apply logarithms to solve for N","content":"To solve for $N$ analytically when it is in the exponent, we take the natural logarithm ($\n\\ln$) of both sides of the inequality:\n\n$$\\ln(1.5^N) > \\ln(313.5)$$\n\nUsing the power rule of logarithms, $\\ln(x^y) = y \\ln(x)$, we can bring the exponent $N$ to the front:\n\n$$N \\ln(1.5) > \\ln(313.5)$$\n\nSince $1.5 > 1$, $\\ln(1.5)$ is a positive number, so we can divide both sides by $\\ln(1.5)$ without reversing the inequality sign:\n\n$$N > \\frac{\\ln(313.5)}{\\ln(1.5)}$$"},{"type":"text","order":4,"title":"Calculate the final month N","content":"Now we evaluate the expression using a calculator:\n\n$$N > 14.172\\dots$$\n\nSince the month $N$ must be a whole number, we find the first integer value of $N$ that satisfies this inequality. \n\nThe first integer greater than $14.17$ is **$15$**.\n\nThus, the total number of premium users is predicted to exceed 5 million for the first time in the **$15$th** month.","calculator":[{"gdc":{"expression":"ln(313.5)/ln(1.5)","instructions":"Enter the logarithmic division formula directly into the run-matrix menu to find the decimal limit."},"ti84":{"expression":"ln(313.5)/ln(1.5)","instructions":"Press [LN], type 313.5, press [)], press [/], press [LN], type 1.5, press [)], then press [ENTER]."},"desmos":{"expression":"\\frac{\\ln(313.5)}{\\ln(1.5)}","instructions":"Type the expression in a blank cell to find the value of N."},"description":"Calculate the exact analytical limit for $N$ using logarithms."}]}]},{"id":"b3bb30d1-85aa-41ba-83b9-c94256e3a632:gdc_table","label":"(c) Calculator Table or Graphical Solver","steps":[{"type":"text","order":0,"title":"Convert 5 million to thousands","content":"Since the number of new premium users is given in **thousands**, we first need to convert 5 million into thousands so that the units of our equation match. \n\n$$\\text{Target} = \\frac{5,000,000}{1,000} = 5000 \\text{ thousand}$$ \n\nWe want to find the smallest integer month $N$ such that the sum of new users up to month $N$ is strictly greater than $5000$:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum formula","content":"We recall the formula for the sum of the first $n$ terms of a geometric series from the formula booklet:\n\n$$S_n = \\frac{a(r^n - 1)}{r - 1}$$\n\nGiven the first term $a = 8$ and the common ratio $r = 1.5$, we substitute these values into the formula to express $S_N$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{1.5 - 1}$$\n\nSimplifying the denominator $1.5 - 1 = 0.5$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{0.5} = 16(1.5^N - 1)$$\n\nSo, our inequality is:\n\n$$16(1.5^N - 1) > 5000$$"},{"type":"text","order":2,"title":"Analyze using calculator table","content":"To find the first integer value of $N$ where $S_N > 5000$, we can use our graphing calculator's table feature to compute $S_N$ for successive values of $N$. Let's define the function:\n\n$$Y_1 = 16(1.5^x - 1)$$\n\nNow, generate the table of values to observe when $Y_1$ first exceeds $5000$.","calculator":[{"gdc":{"expression":"16*(1.5^x - 1)","instructions":"1. Go to the Graph/Table menu.\n2. Enter the function \\\$f1(x) = 16(1.5^x - 1)\\\$.\n3. Open the table of values (ctrl + T on TI-Nspire).\n4. Scroll through the x-values to find when the function value exceeds 5000."},"ti84":{"expression":"16*(1.5^X - 1)","instructions":"1. Press [Y=] and enter \\\$Y_1 = 16(1.5^X - 1)\\\$.\n2. Press [2nd] [WINDOW] (TBLSET). Set TblStart = 10 and \\\$\\Delta\\\$Tbl = 1.\n3. Press [2nd] [GRAPH] (TABLE) to view the table of values.\n4. Scroll down to find the first row where \\\$Y_1 > 5000\\\$."},"desmos":{"expression":"f(x) = 16*(1.5^x - 1)","instructions":"1. Type \\\$f(x) = 16(1.5^x - 1)\\\$ in the input bar.\n2. Click the gear icon (Tools) and select the \"Convert to table\" icon.\n3. Input values of x from 12 to 16 to inspect the corresponding outputs."},"description":"Generate a table of values for the sum function to find when the sum exceeds 5000."}]},{"type":"text","order":3,"title":"Evaluate table values","content":"Using the table of values, we find:\n\n- For $N = 13$:\n $$S_{13} = 16(1.5^{13} - 1) \\approx 16(194.62 - 1) \\approx 3097.9$$\n- For $N = 14$:\n $$S_{14} = 16(1.5^{14} - 1) \\approx 16(291.93 - 1) \\approx 4654.9$$\n- For $N = 15$:\n $$S_{15} = 16(1.5^{15} - 1) \\approx 16(437.89 - 1) \\approx 6990.3$$\n\nThus, the total predicted premium users exceed 5 million ($5000$ thousand) for the first time during month $15$."},{"type":"text","order":4,"title":"State final answer","content":"The first month in which the total number of premium users exceeds 5 million is **15**.\n\n$$N = 15$$"}]}],"generatedAt":"2026-06-09T08:06:23.989Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"9aadfd8a-f1df-4d17-a73d-c82be4a32f66","revisionId":"91883728-ba11-46b5-a1e6-1125d2357938","questionNumber":null,"questionType":"LA","specification":"The arithmetic series \$S\$ is given by\n\n\\[S = 7 + 11 + 15 + 19 + \\cdots + 303\\]\n\nFind","difficulty":3,"difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"21372bfe-f35c-4952-b631-215dba2e31d3","label":"a","content":"the number of terms in the series,","markscheme":"- Identifies the first term $a = 7$, common difference $d = 4$ and last term $l = 303$ B1\n- Uses the formula $u_n = a + (n - 1)d$ to set up the equation $303 = 7 + (n - 1)4$ M1\n- Solves the equation to find $n = 75$ A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"formulaic","label":"Using the n-th Term Formula","steps":[{"type":"text","order":0,"title":"Identify key parameters","content":"To find the number of terms in the arithmetic series, we first need to identify the key components of the sequence:\n\n* **First term ($a$):** The first number in the series is $7$.\n* **Common difference ($d$):** The difference between consecutive terms is $11 - 7 = 4$.\n* **Last term ($u_n$ or $l$):** The final term of the series is $303$.\n\nIdentifying these values correctly earns the first mark B1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$\\htmlData{anchor=un-term}{\\color{@purple}{303}} = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"},{"color":"purple","style":"text-color","anchor":"un-term"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"first term","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"common difference","style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute our identified values into the $n$-th term formula for an arithmetic progression. This substitution earns a method mark M1."},{"type":"text","order":2,"title":"Solve for n","content":"Now we solve the equation for $n$ step-by-step:\n\n$$303 = 7 + 4(n - 1)$$\n\n$$296 = 4(n - 1)$$\n\n$$74 = n - 1$$\n\n$$n = 75$$\n\nThus, there are $75$ terms in the series. Finding this value correctly earns the final mark A1.","calculator":[{"gdc":{"expression":"303 = 7 + (X - 1) * 4","instructions":"Go to the Equation menu, select Solver, enter the equation `303 = 7 + (X - 1) * 4`, and solve for $X$."},"ti84":{"expression":"303 = 7 + (N - 1) * 4","instructions":"Press [MATH], scroll down to [Numeric Solver...]. Enter `E1: 303` and `E2: 7 + (N - 1) * 4`, then press [OK] (graph button) and [SOLVE] to find $N$."},"desmos":{"expression":"303 = 7 + (x - 1) * 4","instructions":"Type the equation in the input bar to find the vertical line representing the root."},"description":"You can verify this result using your graphing calculator's equation solver or by graphing."}]},{"type":"text","order":3,"title":"Calculate the sum","content":"Now we find the sum of the series using the sum formula:\n\n$$S_n = \\frac{n}{2}(a + l)$$\n\nSubstituting $n = 75$, $a = 7$, and $l = 303$ into the formula:\n\n$$S_{75} = \\frac{75}{2}(7 + 303)$$\n\n$$S_{75} = \\frac{75}{2}(310)$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns the method marks M1 M1 and the final accuracy mark A1."}]},{"id":"intuitive","label":"Step-Counting Approach","steps":[{"type":"text","order":0,"title":"Identify key terms","content":"First, let's identify the first term, the last term, and the common difference from the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\cdots + 303$$\n\n* **First term ($a$):** $7$\n* **Last term ($l$):** $303$\n* **Common difference ($d$):** $11 - 7 = 4$\n\n*This earns you your first mark **[B1]**.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Total Difference} = \\htmlData{anchor=last-term}{\\color{@sky}{303}} - \\htmlData{anchor=first-term}{\\color{@amber}{7}} = 296$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"last-term"},{"color":"amber","style":"text-color","anchor":"first-term"}],"connections":[]},"order":1,"title":"Find the total difference","description":"We calculate the total difference between the last term and the first term to see how much the sequence grows overall."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Number of Steps} = \\frac{\\htmlData{anchor=total-diff}{\\color{@teal}{296}}}{\\htmlData{anchor=common-diff}{\\color{@purple}{4}}} = 74$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"total-diff"},{"color":"purple","style":"text-color","anchor":"common-diff"}],"connections":[]},"order":2,"title":"Calculate the steps","description":"Next, divide the total difference by the common difference to find how many steps (or jumps of $4$) are taken to get from the first term to the last term."},{"type":"text","order":3,"title":"Determine total terms","content":"Since we started at the first term, the number of terms is always **one more** than the number of steps:\n\n$$\\text{Number of terms } (n) = \\text{Number of steps} + 1$$\n\n$$n = 74 + 1$$\n\n$$n = 75$$\n\nThis calculation corresponds to solving the standard formula $l = a + (n - 1)d$, which earns the method mark **[M1]** and the final accuracy mark **[A1]**.\n\nTherefore, there are **$75$** terms in the series.","calculator":[{"gdc":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"ti84":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"desmos":{"expression":"\\frac{303-7}{4}+1","instructions":"Enter the calculation into the input field to get the result."},"description":"You can quickly verify the arithmetic on your calculator."}]},{"type":"text","order":4,"title":"Calculate the sum","content":"With the number of terms known, we can calculate the total sum using the alternative arithmetic sum formula:\n\n$$S_n = \\frac{n}{2}[2a + (n - 1)d]$$\n\nSubstituting $n = 75$, $a = 7$, and $d = 4$:\n\n$$S_{75} = \\frac{75}{2}[2(7) + (75 - 1)4]$$\n\n$$S_{75} = \\frac{75}{2}[14 + 74 \\times 4]$$\n\n$$S_{75} = \\frac{75}{2}[310]$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns **[M1]** and the final sum of **$11\\,625$** earns **[A1]**."}]}],"generatedAt":"2026-06-09T07:59:13.284Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523","label":"b","content":"the sum of the series.","markscheme":"- Uses the arithmetic series sum formula $S_n = \\frac{n}{2}(a + l)$ or $S_n = \\frac{n}{2}(2a + (n - 1)d)$ M1\n- Substitutes $n = 75$, $a = 7$, and $l = 303$ into the formula to get $S_{75} = \\frac{75}{2}(7 + 303)$ M1\n- Obtains the final sum $11\\,625$ A1","marks":3,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"first_last_term_formula","label":"Using First and Last Term Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"To sum the arithmetic series $S = 7 + 11 + 15 + 19 + \\dots + 303$, we first identify its key components:\n\n- **First term** ($a$) = $7$\n- **Common difference** ($d$) = $11 - 7 = 4$\n- **Last term** ($l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Use the } n\\text{-th term formula: } \\htmlData{anchor=nth-form}{\\color{@sky}{u_n = a + (n - 1)d}}$"},{"latex":"$$\\htmlData{anchor=sub-eq}{\\color{@sky}{303}} = \\htmlData{anchor=a-sub}{\\color{@purple}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1)$"},{"latex":"$$n - 1 = 74 \\implies \\htmlData{anchor=n-val}{\\color{@green}{n = 75}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-eq"},{"color":"purple","style":"text-color","anchor":"a-sub"},{"color":"amber","style":"text-color","anchor":"d-sub"},{"color":"green","style":"box","anchor":"n-val"}],"connections":[{"to":"sub-eq","from":"nth-form","color":"sky","label":"Substitute values","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We need to determine the total number of terms, $n$, in the series. We use the formula for the $n$-th term of an arithmetic progression, $u_n = a + (n - 1)d$, setting $u_n = 303$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Arithmetic series sum formula: } \\htmlData{anchor=sum-formula}{\\color{@teal}{S_n = \\frac{n}{2}(a + l)}}$"},{"latex":"$$\\htmlData{anchor=sum-sub}{S_{75}} = \\frac{\\htmlData{anchor=n-sub}{\\color{@green}{75}}}{2}\\left(\\htmlData{anchor=a-val}{\\color{@purple}{7}} + \\htmlData{anchor=l-val}{\\color{@sky}{303}}\\right)$"},{"latex":"$$S_{75} = 37.5 \\times 310$"},{"latex":"$$S_{75} = \\htmlData{anchor=final-ans}{\\color{@orange}{11\\,625}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"n-sub"},{"color":"purple","style":"text-color","anchor":"a-val"},{"color":"sky","style":"text-color","anchor":"l-val"},{"color":"orange","style":"box","anchor":"final-ans"}],"connections":[{"to":"sum-sub","from":"sum-formula","color":"teal","label":"Substitute n, a, l","style":"solid"}]},"order":2,"title":"Apply sum formula","description":"Now that we know there are $75$ terms, we can find the sum using the formula $S_n = \\frac{n}{2}(a + l)$, where $l$ is the last term."},{"type":"text","order":3,"title":"Verify with GDC","content":"You can verify this result by evaluating the sum directly using sigma notation on your calculator.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Go to the Run-Matrix menu, press MATH, then Σ(. Input variable X, from 1 to 75, and the expression 7 + (X - 1) × 4."},"ti84":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Press ALPHA then WINDOW (F2), select 2: summation Σ(. Enter the variable X, starting value 1, end value 75, and the formula 7 + (X - 1)*4."},"desmos":{"expression":"\\sum_{n=1}^{75} (7 + (n-1) \\cdot 4)","instructions":"Type sum in an empty expression bar to generate a summation template. Set the limits from n=1 to 75 and input the expression 7 + (n-1)*4."},"description":"Calculate the sum of the series using the sum template."}]}]},{"id":"common_difference_formula","label":"Using Common Difference Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"From the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nWe can identify the key parameters:\n\n- First term ($a$) = $7$\n- Common difference ($d$) = $11 - 7 = 4$\n- Last term ($u_n$ or $l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$303 = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1) \\implies n - 1 = 74 \\implies n = 75$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"substitute a = 7","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"substitute d = 4","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We use the general $n^{\\text{th}}$ term formula for an arithmetic sequence to solve for $n$, the total number of terms in the series."},{"type":"text","order":2,"title":"Calculate the sum","content":"Now we substitute our values ($n = 75$, $a = 7$, and $d = 4$) into the arithmetic series sum formula involving the common difference:\n\n$$S_n = \\frac{n}{2}\\left(2a + (n - 1)d\\right)$$\n\n$$\\begin{aligned} S_{75} &= \\frac{75}{2}\\left(2(7) + (75 - 1)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + (74)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + 296\\right) \\\\ &= \\frac{75}{2}(310) \\\\ &= 75 \\times 155 \\\\ &= 11\\,625 \\end{aligned}$$\n\nThis earns the method marks M1 M1 and accuracy mark A1.","calculator":[{"gdc":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Enter '75 / 2 * (2 * 7 + 74 * 4)' on the run screen and execute."},"ti84":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Type the arithmetic expression on the home screen: '75 / 2 * (2 * 7 + 74 * 4)' and press ENTER."},"desmos":{"expression":"\\frac{75}{2}\\left(2\\cdot7+\\left(75-1\\right)\\cdot4\\right)","instructions":"Type the formula directly to evaluate the sum."},"description":"Verify the calculation on your graphing calculator."}]}]},{"id":"calculator_sigma_notation","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Analyze the series","content":"The given arithmetic series is:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nTo represent this series in sigma notation, we first find the general term ($r$-th term), $u_r$.\n\n- The first term is $a = 7$.\n- The common difference is $d = 11 - 7 = 4$.\n\nThe formula for the $r$-th term of an arithmetic sequence is:\n\n$$u_r = a + (r - 1)d$$\n\nSubstituting our values in:\n\n$$u_r = 7 + (r - 1)4 = 4r + 3$$"},{"type":"text","order":1,"title":"Find number of terms","content":"Next, we find the number of terms, $n$, by setting the general term equal to the last term, $303$:\n\n$$4n + 3 = 303$$\n\nSolving for $n$:\n\n$$\\begin{aligned} 4n &= 300 \\\\ n &= 75 \\end{aligned}$$\n\nSo, there are $75$ terms in this series."},{"type":"text","order":2,"title":"Write in sigma notation","content":"We can now write the sum of the series using sigma notation as:\n\n$$S = \\sum_{r=1}^{75} (4r + 3)$$\n\nWe can enter this expression directly into a graphing calculator to find the final sum."},{"type":"text","order":3,"title":"Evaluate with calculator","content":"Inputting the summation into your graphing calculator will evaluate the total sum of the series directly.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (4X+3)","instructions":"Go to the Run-Matrix menu. Press [MATH] then [F6] to see more options, and select Sigma ([F2]).\nFill in the template:\n- Enter the expression: 4X+3\n- Set the variable: X\n- Set the start value: 1\n- Set the end value: 75\nPress [EXE] to calculate."},"ti84":{"expression":"sum(seq(4X+3,X,1,75))","instructions":"Press [ALPHA] [WINDOW] and select 2:summation(\nOr press [MATH] and scroll down to 0:summation(\nFill in the template:\n- Under the Sigma symbol, enter X=1\n- On top of the Sigma symbol, enter 75\n- In the parentheses, enter 4X+3\nPress [ENTER] to calculate."},"desmos":{"expression":"\\sum_{n=1}^{75} (4n+3)","instructions":"Type 'sum' in an expression line to bring up the summation notation.\nSet the lower limit to n=1, the upper limit to 75, and enter 4n+3 inside the summation."},"description":"Evaluate the sum of 4r + 3 from r = 1 to 75."}]},{"type":"text","order":4,"title":"State final answer","content":"The graphing calculator outputs the final sum:\n\n$$S = 11\\,625$$\n\nThis confirms the analytical calculation of the arithmetic series sum."}]}],"generatedAt":"2026-06-09T08:00:56.446Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"21372bfe-f35c-4952-b631-215dba2e31d3:formulaic","label":"(a) Using the n-th Term Formula","steps":[{"type":"text","order":0,"title":"Identify key parameters","content":"To find the number of terms in the arithmetic series, we first need to identify the key components of the sequence:\n\n* **First term ($a$):** The first number in the series is $7$.\n* **Common difference ($d$):** The difference between consecutive terms is $11 - 7 = 4$.\n* **Last term ($u_n$ or $l$):** The final term of the series is $303$.\n\nIdentifying these values correctly earns the first mark B1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$\\htmlData{anchor=un-term}{\\color{@purple}{303}} = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"},{"color":"purple","style":"text-color","anchor":"un-term"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"first term","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"common difference","style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute our identified values into the $n$-th term formula for an arithmetic progression. This substitution earns a method mark M1."},{"type":"text","order":2,"title":"Solve for n","content":"Now we solve the equation for $n$ step-by-step:\n\n$$303 = 7 + 4(n - 1)$$\n\n$$296 = 4(n - 1)$$\n\n$$74 = n - 1$$\n\n$$n = 75$$\n\nThus, there are $75$ terms in the series. Finding this value correctly earns the final mark A1.","calculator":[{"gdc":{"expression":"303 = 7 + (X - 1) * 4","instructions":"Go to the Equation menu, select Solver, enter the equation `303 = 7 + (X - 1) * 4`, and solve for $X$."},"ti84":{"expression":"303 = 7 + (N - 1) * 4","instructions":"Press [MATH], scroll down to [Numeric Solver...]. Enter `E1: 303` and `E2: 7 + (N - 1) * 4`, then press [OK] (graph button) and [SOLVE] to find $N$."},"desmos":{"expression":"303 = 7 + (x - 1) * 4","instructions":"Type the equation in the input bar to find the vertical line representing the root."},"description":"You can verify this result using your graphing calculator's equation solver or by graphing."}]},{"type":"text","order":3,"title":"Calculate the sum","content":"Now we find the sum of the series using the sum formula:\n\n$$S_n = \\frac{n}{2}(a + l)$$\n\nSubstituting $n = 75$, $a = 7$, and $l = 303$ into the formula:\n\n$$S_{75} = \\frac{75}{2}(7 + 303)$$\n\n$$S_{75} = \\frac{75}{2}(310)$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns the method marks M1 M1 and the final accuracy mark A1."}]},{"id":"21372bfe-f35c-4952-b631-215dba2e31d3:intuitive","label":"(a) Step-Counting Approach","steps":[{"type":"text","order":0,"title":"Identify key terms","content":"First, let's identify the first term, the last term, and the common difference from the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\cdots + 303$$\n\n* **First term ($a$):** $7$\n* **Last term ($l$):** $303$\n* **Common difference ($d$):** $11 - 7 = 4$\n\n*This earns you your first mark **[B1]**.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Total Difference} = \\htmlData{anchor=last-term}{\\color{@sky}{303}} - \\htmlData{anchor=first-term}{\\color{@amber}{7}} = 296$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"last-term"},{"color":"amber","style":"text-color","anchor":"first-term"}],"connections":[]},"order":1,"title":"Find the total difference","description":"We calculate the total difference between the last term and the first term to see how much the sequence grows overall."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Number of Steps} = \\frac{\\htmlData{anchor=total-diff}{\\color{@teal}{296}}}{\\htmlData{anchor=common-diff}{\\color{@purple}{4}}} = 74$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"total-diff"},{"color":"purple","style":"text-color","anchor":"common-diff"}],"connections":[]},"order":2,"title":"Calculate the steps","description":"Next, divide the total difference by the common difference to find how many steps (or jumps of $4$) are taken to get from the first term to the last term."},{"type":"text","order":3,"title":"Determine total terms","content":"Since we started at the first term, the number of terms is always **one more** than the number of steps:\n\n$$\\text{Number of terms } (n) = \\text{Number of steps} + 1$$\n\n$$n = 74 + 1$$\n\n$$n = 75$$\n\nThis calculation corresponds to solving the standard formula $l = a + (n - 1)d$, which earns the method mark **[M1]** and the final accuracy mark **[A1]**.\n\nTherefore, there are **$75$** terms in the series.","calculator":[{"gdc":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"ti84":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"desmos":{"expression":"\\frac{303-7}{4}+1","instructions":"Enter the calculation into the input field to get the result."},"description":"You can quickly verify the arithmetic on your calculator."}]},{"type":"text","order":4,"title":"Calculate the sum","content":"With the number of terms known, we can calculate the total sum using the alternative arithmetic sum formula:\n\n$$S_n = \\frac{n}{2}[2a + (n - 1)d]$$\n\nSubstituting $n = 75$, $a = 7$, and $d = 4$:\n\n$$S_{75} = \\frac{75}{2}[2(7) + (75 - 1)4]$$\n\n$$S_{75} = \\frac{75}{2}[14 + 74 \\times 4]$$\n\n$$S_{75} = \\frac{75}{2}[310]$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns **[M1]** and the final sum of **$11\\,625$** earns **[A1]**."}]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523:first_last_term_formula","label":"(b) Using First and Last Term Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"To sum the arithmetic series $S = 7 + 11 + 15 + 19 + \\dots + 303$, we first identify its key components:\n\n- **First term** ($a$) = $7$\n- **Common difference** ($d$) = $11 - 7 = 4$\n- **Last term** ($l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Use the } n\\text{-th term formula: } \\htmlData{anchor=nth-form}{\\color{@sky}{u_n = a + (n - 1)d}}$"},{"latex":"$$\\htmlData{anchor=sub-eq}{\\color{@sky}{303}} = \\htmlData{anchor=a-sub}{\\color{@purple}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1)$"},{"latex":"$$n - 1 = 74 \\implies \\htmlData{anchor=n-val}{\\color{@green}{n = 75}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-eq"},{"color":"purple","style":"text-color","anchor":"a-sub"},{"color":"amber","style":"text-color","anchor":"d-sub"},{"color":"green","style":"box","anchor":"n-val"}],"connections":[{"to":"sub-eq","from":"nth-form","color":"sky","label":"Substitute values","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We need to determine the total number of terms, $n$, in the series. We use the formula for the $n$-th term of an arithmetic progression, $u_n = a + (n - 1)d$, setting $u_n = 303$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Arithmetic series sum formula: } \\htmlData{anchor=sum-formula}{\\color{@teal}{S_n = \\frac{n}{2}(a + l)}}$"},{"latex":"$$\\htmlData{anchor=sum-sub}{S_{75}} = \\frac{\\htmlData{anchor=n-sub}{\\color{@green}{75}}}{2}\\left(\\htmlData{anchor=a-val}{\\color{@purple}{7}} + \\htmlData{anchor=l-val}{\\color{@sky}{303}}\\right)$"},{"latex":"$$S_{75} = 37.5 \\times 310$"},{"latex":"$$S_{75} = \\htmlData{anchor=final-ans}{\\color{@orange}{11\\,625}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"n-sub"},{"color":"purple","style":"text-color","anchor":"a-val"},{"color":"sky","style":"text-color","anchor":"l-val"},{"color":"orange","style":"box","anchor":"final-ans"}],"connections":[{"to":"sum-sub","from":"sum-formula","color":"teal","label":"Substitute n, a, l","style":"solid"}]},"order":2,"title":"Apply sum formula","description":"Now that we know there are $75$ terms, we can find the sum using the formula $S_n = \\frac{n}{2}(a + l)$, where $l$ is the last term."},{"type":"text","order":3,"title":"Verify with GDC","content":"You can verify this result by evaluating the sum directly using sigma notation on your calculator.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Go to the Run-Matrix menu, press MATH, then Σ(. Input variable X, from 1 to 75, and the expression 7 + (X - 1) × 4."},"ti84":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Press ALPHA then WINDOW (F2), select 2: summation Σ(. Enter the variable X, starting value 1, end value 75, and the formula 7 + (X - 1)*4."},"desmos":{"expression":"\\sum_{n=1}^{75} (7 + (n-1) \\cdot 4)","instructions":"Type sum in an empty expression bar to generate a summation template. Set the limits from n=1 to 75 and input the expression 7 + (n-1)*4."},"description":"Calculate the sum of the series using the sum template."}]}]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523:common_difference_formula","label":"(b) Using Common Difference Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"From the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nWe can identify the key parameters:\n\n- First term ($a$) = $7$\n- Common difference ($d$) = $11 - 7 = 4$\n- Last term ($u_n$ or $l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$303 = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1) \\implies n - 1 = 74 \\implies n = 75$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"substitute a = 7","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"substitute d = 4","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We use the general $n^{\\text{th}}$ term formula for an arithmetic sequence to solve for $n$, the total number of terms in the series."},{"type":"text","order":2,"title":"Calculate the sum","content":"Now we substitute our values ($n = 75$, $a = 7$, and $d = 4$) into the arithmetic series sum formula involving the common difference:\n\n$$S_n = \\frac{n}{2}\\left(2a + (n - 1)d\\right)$$\n\n$$\\begin{aligned} S_{75} &= \\frac{75}{2}\\left(2(7) + (75 - 1)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + (74)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + 296\\right) \\\\ &= \\frac{75}{2}(310) \\\\ &= 75 \\times 155 \\\\ &= 11\\,625 \\end{aligned}$$\n\nThis earns the method marks M1 M1 and accuracy mark A1.","calculator":[{"gdc":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Enter '75 / 2 * (2 * 7 + 74 * 4)' on the run screen and execute."},"ti84":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Type the arithmetic expression on the home screen: '75 / 2 * (2 * 7 + 74 * 4)' and press ENTER."},"desmos":{"expression":"\\frac{75}{2}\\left(2\\cdot7+\\left(75-1\\right)\\cdot4\\right)","instructions":"Type the formula directly to evaluate the sum."},"description":"Verify the calculation on your graphing calculator."}]}]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523:calculator_sigma_notation","label":"(b) Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Analyze the series","content":"The given arithmetic series is:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nTo represent this series in sigma notation, we first find the general term ($r$-th term), $u_r$.\n\n- The first term is $a = 7$.\n- The common difference is $d = 11 - 7 = 4$.\n\nThe formula for the $r$-th term of an arithmetic sequence is:\n\n$$u_r = a + (r - 1)d$$\n\nSubstituting our values in:\n\n$$u_r = 7 + (r - 1)4 = 4r + 3$$"},{"type":"text","order":1,"title":"Find number of terms","content":"Next, we find the number of terms, $n$, by setting the general term equal to the last term, $303$:\n\n$$4n + 3 = 303$$\n\nSolving for $n$:\n\n$$\\begin{aligned} 4n &= 300 \\\\ n &= 75 \\end{aligned}$$\n\nSo, there are $75$ terms in this series."},{"type":"text","order":2,"title":"Write in sigma notation","content":"We can now write the sum of the series using sigma notation as:\n\n$$S = \\sum_{r=1}^{75} (4r + 3)$$\n\nWe can enter this expression directly into a graphing calculator to find the final sum."},{"type":"text","order":3,"title":"Evaluate with calculator","content":"Inputting the summation into your graphing calculator will evaluate the total sum of the series directly.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (4X+3)","instructions":"Go to the Run-Matrix menu. Press [MATH] then [F6] to see more options, and select Sigma ([F2]).\nFill in the template:\n- Enter the expression: 4X+3\n- Set the variable: X\n- Set the start value: 1\n- Set the end value: 75\nPress [EXE] to calculate."},"ti84":{"expression":"sum(seq(4X+3,X,1,75))","instructions":"Press [ALPHA] [WINDOW] and select 2:summation(\nOr press [MATH] and scroll down to 0:summation(\nFill in the template:\n- Under the Sigma symbol, enter X=1\n- On top of the Sigma symbol, enter 75\n- In the parentheses, enter 4X+3\nPress [ENTER] to calculate."},"desmos":{"expression":"\\sum_{n=1}^{75} (4n+3)","instructions":"Type 'sum' in an expression line to bring up the summation notation.\nSet the lower limit to n=1, the upper limit to 75, and enter 4n+3 inside the summation."},"description":"Evaluate the sum of 4r + 3 from r = 1 to 75."}]},{"type":"text","order":4,"title":"State final answer","content":"The graphing calculator outputs the final sum:\n\n$$S = 11\\,625$$\n\nThis confirms the analytical calculation of the arithmetic series sum."}]}],"generatedAt":"2026-06-09T08:01:10.012Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"9ae90cf4-e801-4c54-90f0-26be65141b25","revisionId":"eb3bb87e-83d0-44b4-9a22-395ccd69c69a","questionNumber":null,"questionType":"LA","specification":"A series of robotic pulses is programmed such that the energy, \$ E_k \$, of the \$ k \$-th pulse follows a geometric sequence with first term \$ a \$ Joules and common ratio \$ r \$, where \$ r > 0 \$.\n\nGiven that\n- the 2nd pulse has energy \$ 108 \$ Joules\n- the 4th pulse has energy \$ 48 \$ Joules","difficulty":4,"difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"6cfec091-2f66-4aca-9705-769237196435","label":"a","content":"Show that \$ r = \\frac{2}{3} \$.","markscheme":"- Uses $E_2 = ar = 108$ and $E_4 = ar^3 = 48$ to write $\\frac{ar^3}{ar} = \\frac{48}{108}$ M1\n- Simplifies to $r^2 = \\frac{4}{9}$ and concludes $r = \\frac{2}{3}$ since $r > 0$ A1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"f8005a32-51f7-4762-8a45-33ccb4217f5a","label":"b","content":"Determine the value of \$ a \$.","markscheme":"- Substitutes $r = \\frac{2}{3}$ into $ar = 108$ to find $a = 162$ B1","marks":1,"order":1,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"816d73c3-9983-4c01-b039-5fe860d16e24","label":"c","content":"Given that the total energy released by the first \$ n \$ pulses is greater than \$ 480 \$ Joules,\n\nfind the smallest possible value of \$ n \$.\n\n(Solutions based entirely on graphical or numerical methods are not acceptable.)","markscheme":"- Substitutes $a = 162$ and $r = \\frac{2}{3}$ into the sum formula $S_n = \\frac{a(1 - r^n)}{1 - r}$ to set up the inequality $\\frac{162(1 - (2/3)^n)}{1 - 2/3} > 480$ M1\n- Simplifies to $486(1 - (2/3)^n) > 480$, leading to $(2/3)^n < \\frac{6}{486}$ or $(2/3)^n < \\frac{1}{81}$ A1\n- Applies logarithms to solve for $n$, e.g. $n \\ln(2/3) < \\ln(1/81)$ or $n > \\frac{\\ln(1/81)}{\\ln(2/3)}$ M1\n- Obtains $n > 10.838...$ and concludes the smallest integer value is $n = 11$ 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Sequences and series (A-level only)","topicId":"053492e1-c402-4158-af07-bb52b29684d7","topicSidebar":"$L3c","topicSlug":"sequences-and-series-4","topicTitle":"3.3 Sequences and series (A-level only)"}]}],"$L3d","$L3e"]}]

3.3 Sequences and series (A-level only)

Question 109

3.3 Sequences and series (A-level only) Questions

3.3 Sequences and series (A-level only)

Question 109

3.3 Sequences and series (A-level only) Questions