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Recurrence Relations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"386d08df-0eaa-450c-9b00-c51c22fd48e3","label":"a","content":"Find an expression for $x_2$ in terms of $k$.","markscheme":"- Substitutes $x_1 = k$ into the relation: $x_2 = 4k - 1$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct expression $4k - 1$.","code":"B"}],"finalAnswers":["4k - 1"],"generatedAt":"2026-05-06T03:40:19.006Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_substitution","label":"Direct Substitution","steps":[{"type":"text","order":0,"title":"Identify the relation","content":"We are given a sequence defined by a recurrence relation:\n\n$$x_n = 4x_{n-1} - 1 \\quad \\text{for } n \\geq 2$$\n\nwith the first term given as:\n\n$$x_1 = k$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=sub-n}{x_2} = 4\\htmlData{anchor=sub-n-minus}{x_{2-1}} - 1$"},{"latex":"$$x_2 = 4\\htmlData{anchor=term-prev}{x_1} - 1$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-n"}],"connections":[{"to":"sub-n-minus","from":"sub-n","color":"sky","label":"n = 2","style":"dashed"}]},"order":1,"title":"Substitute n = 2","description":"To find $x_2$, we substitute $n = 2$ directly into the recurrence relation."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x_1 = \\htmlData{anchor=val-k}{k}$"},{"latex":"$$x_2 = 4\\htmlData{anchor=sub-k}{x_1} - 1$"},{"latex":"$$x_2 = 4k - 1$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"val-k"},{"color":"purple","style":"text-color","anchor":"sub-k"}],"connections":[{"to":"sub-k","from":"val-k","color":"purple","label":"Substitute k","style":"solid"}]},"order":2,"title":"Substitute x_1 = k","description":"Now, substitute the value of the first term, $x_1 = k$, into the expression to write $x_2$ in terms of $k$."},{"type":"text","order":3,"title":"Find x_3","content":"To show that $x_3 = 16k - 5$, substitute our expression for $x_2$ back into the recurrence relation for $n = 3$:\n\n$$x_3 = 4x_2 - 1$$\n\nSubstitute $x_2 = 4k - 1$:\n\n$$x_3 = 4(4k - 1) - 1$$\n\nExpand and simplify:\n\n$$x_3 = 16k - 4 - 1 = 16k - 5$$"},{"type":"text","order":4,"title":"Find x_4","content":"To find the sum of the first four terms, we first need to determine the expression for $x_4$. Using the recurrence relation with $n = 4$:\n\n$$x_4 = 4x_3 - 1$$\n\nSubstitute our expression for $x_3$:\n\n$$x_4 = 4(16k - 5) - 1$$\n\nExpand and simplify:\n\n$$x_4 = 64k - 20 - 1 = 64k - 21$$"},{"type":"text","order":5,"title":"Calculate the sum","content":"Now, sum the first four terms of the sequence, $\\sum_{r=1}^4 x_r$:\n\n$$\\sum_{r=1}^4 x_r = x_1 + x_2 + x_3 + x_4$$\n\nSubstitute the expressions in terms of $k$:\n\n$$\\sum_{r=1}^4 x_r = k + (4k - 1) + (16k - 5) + (64k - 21)$$\n\nGroup the like terms:\n\n$$\\sum_{r=1}^4 x_r = (1 + 4 + 16 + 64)k - (1 + 5 + 21)$$\n\n$$\\sum_{r=1}^4 x_r = 85k - 27$$"}]}],"generatedAt":"2026-06-09T09:59:01.623Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"e7ab2bcf-0161-4664-ab05-742af196bb18","label":"b","content":"Show that $x_3 = 16k - 5$.","markscheme":"- Substitutes $x_2$ into the relation: $x_3 = 4(4k - 1) - 1$ M1\n- Simplifies to $16k - 4 - 1 = 16k - 5$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Attempts to find $x_3$ by substituting $x_2$ into the recurrence relation.","code":"M"},{"id":"p2","marks":1,"description":"Correctly simplifies to reach $16k - 5$.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T03:40:19.006Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_substitution","label":"Direct Substitution","steps":[{"type":"text","order":0,"title":"Find the second term","content":"To find $x_3$, we first need to determine the expression for $x_2$. \n\nWe are given the recurrence relation:\n$$x_n = 4x_{n-1} - 1$$\n\nFor $n = 2$, this formula gives:\n$$x_2 = 4x_1 - 1$$\n\nSubstitute the given initial term $x_1 = k$ into the formula:\n$$x_2 = 4k - 1$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x_2 = \\htmlData{anchor=x2-val}{\\color{@sky}{4k - 1}}$"},{"latex":"$$$\\begin{aligned} x_3 &= 4\\htmlData{anchor=x2-sub}{\\color{@sky}{x_2}} - 1 \\\\ &= 4\\htmlData{anchor=sub-target}{\\color{@sky}{\\left(4k - 1\\right)}} - 1 \\end{aligned}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x2-val"},{"color":"sky","style":"text-color","anchor":"sub-target"}],"connections":[{"to":"sub-target","from":"x2-val","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Substitute into relation","description":"Next, to find $x_3$, we substitute our expression for $x_2$ back into the recurrence relation for $n = 3$. This step earns the first method mark (M1)."},{"type":"text","order":2,"title":"Simplify the expression","content":"Now, expand the brackets and simplify the equation to find the final expression for $x_3$:\n\n$$\\begin{aligned} x_3 &= 4(4k - 1) - 1 \\\\ &= 16k - 4 - 1 \\\\ &= 16k - 5 \\end{aligned}$$\n\nThis final simplification completes the proof and earns the second mark (A1)."}]}],"generatedAt":"2026-06-11T03:20:44.008Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"087e4553-a31a-4cb1-9468-ffd82c8349ae","label":"c","content":"Find $\\sum_{r=1}^4 x_r$ in terms of $k$.","markscheme":"- Finds $x_4 = 4(16k - 5) - 1 = 64k - 21$ M1\n- Sums the four terms: $k + (4k - 1) + (16k - 5) + (64k - 21)$ M1\n- Simplifies to $85k - 27$ A1","marks":3,"order":2,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly finds the expression for $x_4$.","code":"M"},{"id":"p2","marks":1,"description":"Attempts to sum all four terms $x_1 + x_2 + x_3 + x_4$.","code":"M"},{"id":"p3","marks":1,"description":"Final simplified expression $85k - 27$.","code":"A"}],"finalAnswers":["85k - 27"],"generatedAt":"2026-05-06T03:40:19.006Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"term_by_term","label":"Term-by-Term Calculation","steps":[{"type":"text","order":0,"title":"Understand the summation task","content":"We need to find the value of the sum of the first four terms of our sequence, $\\sum_{r=1}^4 x_r$, in terms of $k$:\n\n$$\\sum_{r=1}^4 x_r = x_1 + x_2 + x_3 + x_4$$\n\nTo do this, we will find each individual term starting from $x_1$ up to $x_4$ using the recurrence relation, and then add them together.","highlights":[{"text":"Find $\\sum_{r=1}^4 x_r$ in terms of $k$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=x1-val}{\\color{@sky}{x_1 = k}}$"},{"latex":"$$\\htmlData{anchor=x2-val}{\\color{@amber}{x_2}} = 4(\\color{@sky}{x_1}) - 1 = \\htmlData{anchor=x2-expr}{\\color{@amber}{4k - 1}}$"},{"latex":"$$\\htmlData{anchor=x3-val}{\\color{@purple}{x_3}} = 4(\\color{@amber}{x_2}) - 1 = 4(\\color{@amber}{4k - 1}) - 1 = \\htmlData{anchor=x3-expr}{\\color{@purple}{16k - 5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x1-val"},{"color":"amber","style":"text-color","anchor":"x2-expr"},{"color":"purple","style":"text-color","anchor":"x3-expr"}],"connections":[{"to":"x2-val","from":"x1-val","color":"sky","label":"substitute x1","style":"solid"},{"to":"x3-val","from":"x2-expr","color":"amber","label":"substitute x2","style":"solid"}]},"order":1,"title":"Find the first three terms","description":"Using the relation $x_n = 4x_{n-1} - 1$, we can determine the first three terms step-by-step in terms of $k$."},{"type":"text","order":2,"title":"Calculate the fourth term","content":"Now we calculate the fourth term, $x_4$, by substituting our expression for $x_3 = 16k - 5$ into the recurrence relation:\n\n$$\\begin{aligned} x_4 &= 4x_3 - 1 \\\\ &= 4(16k - 5) - 1 \\\\ &= 64k - 20 - 1 \\\\ &= 64k - 21 \\end{aligned}$$\n\nThis calculation earns you the first method mark (**M1**)."},{"type":"text","order":3,"title":"Sum the four terms","content":"Now, we write down the sum of the first four terms:\n\n$$\\sum_{r=1}^4 x_r = x_1 + x_2 + x_3 + x_4$$\n\nSubstituting our expressions for each term, we get:\n\n$$\\sum_{r=1}^4 x_r = (k) + (4k - 1) + (16k - 5) + (64k - 21)$$\n\nWriting this sum earns the second method mark (**M1**)."},{"type":"text","order":4,"title":"Simplify the expression","content":"To get our final answer, we group the $k$ terms and the constant terms together:\n\n$$\\begin{aligned} \\sum_{r=1}^4 x_r &= (1 + 4 + 16 + 64)k + (-1 - 5 - 21) \\\\ &= 85k - 27 \\end{aligned}$$\n\nThis simplified final result earns the accuracy mark (**A1**)."}]},{"id":"closed_form_gp","label":"Finding the General Term","steps":[{"type":"text","order":0,"title":"Analyze the recurrence relation","content":"We are given the first-order linear recurrence relation:\n\n$$x_n = 4x_{n-1} - 1 \\quad \\text{for } n \\geq 2$$ \n\nwith initial term $x_1 = k$. To find the sum $\\sum_{r=1}^4 x_r$, we will first find a closed-form expression for the general term $x_n$ by representing it as a shifted geometric sequence."},{"type":"text","order":1,"title":"Find the constant shift","content":"Let's find a constant $c$ such that the sequence $y_n = x_n - c$ is geometric. If $y_n$ is geometric with a common ratio of $4$, then:\n\n$$y_n = 4y_{n-1}$$\n\nSubstituting $y_n = x_n - c$ into this relation gives:\n\n$$x_n - c = 4(x_{n-1} - c)$$\n\n$$x_n = 4x_{n-1} - 3c$$\n\nComparing this with our original recurrence relation $x_n = 4x_{n-1} - 1$, we get:\n\n$$-3c = -1 \\implies c = \\frac{1}{3}$$"},{"type":"text","order":2,"title":"Define the shifted sequence","content":"Using our value of $c = \\frac{1}{3}$, we can define the geometric sequence $y_n$ as:\n\n$$y_n = x_n - \\frac{1}{3}$$\n\nLet's find the first term $y_1$:\n\n$$y_1 = x_1 - \\frac{1}{3} = k - \\frac{1}{3}$$\n\nSince $y_n$ has a common ratio $r = 4$, the general term $y_n$ is:\n\n$$y_n = y_1 \\cdot r^{n-1} = \\left(k - \\frac{1}{3}\\right) 4^{n-1}$$"},{"type":"text","order":3,"title":"Express the general term","content":"Now we can write the general term $x_n$ in closed form by adding the shift back:\n\n$$x_n = y_n + \\frac{1}{3} = \\left(k - \\frac{1}{3}\\right) 4^{n-1} + \\frac{1}{3}$$\n\nLet's verify this formula for $n = 1$:\n\n$$x_1 = \\left(k - \\frac{1}{3}\\right) 4^0 + \\frac{1}{3} = k$$"},{"type":"text","order":4,"title":"Sum the sequence","content":"We need to find the sum of the first four terms, $\\sum_{r=1}^4 x_r$. Using our general term expression:\n\n$$\\sum_{r=1}^4 x_r = \\sum_{r=1}^4 \\left[ \\left(k - \\frac{1}{3}\\right) 4^{r-1} + \\frac{1}{3} \\right]$$\n\nWe can split this into two parts:\n\n$$\\sum_{r=1}^4 x_r = \\left(k - \\frac{1}{3}\\right) \\sum_{r=1}^4 4^{r-1} + \\sum_{r=1}^4 \\frac{1}{3}$$"},{"type":"text","order":5,"title":"Calculate the sums","content":"First, evaluate the geometric sum $\\sum_{r=1}^4 4^{r-1}$ using the geometric series sum formula $S_n = \\frac{a(q^n - 1)}{q - 1}$ where $a = 1, q = 4, n = 4$:\n\n$$\\sum_{r=1}^4 4^{r-1} = \\frac{1(4^4 - 1)}{4 - 1} = \\frac{256 - 1}{3} = \\frac{255}{3} = 85$$\n\nNext, sum the constant term $\\frac{1}{3}$ over $4$ terms:\n\n$$\\sum_{r=1}^4 \\frac{1}{3} = 4 \\times \\frac{1}{3} = \\frac{4}{3}$$"},{"type":"text","order":6,"title":"Combine and simplify","content":"Now substitute these values back into our sum expression:\n\n$$\\begin{aligned} \\sum_{r=1}^4 x_r &= 85\\left(k - \\frac{1}{3}\\right) + \\frac{4}{3} \\\\ &= 85k - \\frac{85}{3} + \\frac{4}{3} \\\\ &= 85k - \\frac{81}{3} \\\\ &= 85k - 27 \\end{aligned}$$\n\nThis matches the simplified sum of the individual terms: $k + (4k-1) + (16k-5) + (64k-21) = 85k-27$."}]}],"generatedAt":"2026-06-11T03:21:03.111Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"386d08df-0eaa-450c-9b00-c51c22fd48e3:direct_substitution","label":"(a) Direct Substitution","steps":[{"type":"text","order":0,"title":"Identify the relation","content":"We are given a sequence defined by a recurrence relation:\n\n$$x_n = 4x_{n-1} - 1 \\quad \\text{for } n \\geq 2$$\n\nwith the first term given as:\n\n$$x_1 = k$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=sub-n}{x_2} = 4\\htmlData{anchor=sub-n-minus}{x_{2-1}} - 1$"},{"latex":"$$x_2 = 4\\htmlData{anchor=term-prev}{x_1} - 1$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-n"}],"connections":[{"to":"sub-n-minus","from":"sub-n","color":"sky","label":"n = 2","style":"dashed"}]},"order":1,"title":"Substitute n = 2","description":"To find $x_2$, we substitute $n = 2$ directly into the recurrence relation."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x_1 = \\htmlData{anchor=val-k}{k}$"},{"latex":"$$x_2 = 4\\htmlData{anchor=sub-k}{x_1} - 1$"},{"latex":"$$x_2 = 4k - 1$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"val-k"},{"color":"purple","style":"text-color","anchor":"sub-k"}],"connections":[{"to":"sub-k","from":"val-k","color":"purple","label":"Substitute k","style":"solid"}]},"order":2,"title":"Substitute x_1 = k","description":"Now, substitute the value of the first term, $x_1 = k$, into the expression to write $x_2$ in terms of $k$."},{"type":"text","order":3,"title":"Find x_3","content":"To show that $x_3 = 16k - 5$, substitute our expression for $x_2$ back into the recurrence relation for $n = 3$:\n\n$$x_3 = 4x_2 - 1$$\n\nSubstitute $x_2 = 4k - 1$:\n\n$$x_3 = 4(4k - 1) - 1$$\n\nExpand and simplify:\n\n$$x_3 = 16k - 4 - 1 = 16k - 5$$"},{"type":"text","order":4,"title":"Find x_4","content":"To find the sum of the first four terms, we first need to determine the expression for $x_4$. Using the recurrence relation with $n = 4$:\n\n$$x_4 = 4x_3 - 1$$\n\nSubstitute our expression for $x_3$:\n\n$$x_4 = 4(16k - 5) - 1$$\n\nExpand and simplify:\n\n$$x_4 = 64k - 20 - 1 = 64k - 21$$"},{"type":"text","order":5,"title":"Calculate the sum","content":"Now, sum the first four terms of the sequence, $\\sum_{r=1}^4 x_r$:\n\n$$\\sum_{r=1}^4 x_r = x_1 + x_2 + x_3 + x_4$$\n\nSubstitute the expressions in terms of $k$:\n\n$$\\sum_{r=1}^4 x_r = k + (4k - 1) + (16k - 5) + (64k - 21)$$\n\nGroup the like terms:\n\n$$\\sum_{r=1}^4 x_r = (1 + 4 + 16 + 64)k - (1 + 5 + 21)$$\n\n$$\\sum_{r=1}^4 x_r = 85k - 27$$"}]},{"id":"e7ab2bcf-0161-4664-ab05-742af196bb18:direct_substitution","label":"(b) Direct Substitution","steps":[{"type":"text","order":0,"title":"Find the second term","content":"To find $x_3$, we first need to determine the expression for $x_2$. \n\nWe are given the recurrence relation:\n$$x_n = 4x_{n-1} - 1$$\n\nFor $n = 2$, this formula gives:\n$$x_2 = 4x_1 - 1$$\n\nSubstitute the given initial term $x_1 = k$ into the formula:\n$$x_2 = 4k - 1$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x_2 = \\htmlData{anchor=x2-val}{\\color{@sky}{4k - 1}}$"},{"latex":"$$$\\begin{aligned} x_3 &= 4\\htmlData{anchor=x2-sub}{\\color{@sky}{x_2}} - 1 \\\\ &= 4\\htmlData{anchor=sub-target}{\\color{@sky}{\\left(4k - 1\\right)}} - 1 \\end{aligned}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x2-val"},{"color":"sky","style":"text-color","anchor":"sub-target"}],"connections":[{"to":"sub-target","from":"x2-val","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Substitute into relation","description":"Next, to find $x_3$, we substitute our expression for $x_2$ back into the recurrence relation for $n = 3$. This step earns the first method mark (M1)."},{"type":"text","order":2,"title":"Simplify the expression","content":"Now, expand the brackets and simplify the equation to find the final expression for $x_3$:\n\n$$\\begin{aligned} x_3 &= 4(4k - 1) - 1 \\\\ &= 16k - 4 - 1 \\\\ &= 16k - 5 \\end{aligned}$$\n\nThis final simplification completes the proof and earns the second mark (A1)."}]},{"id":"087e4553-a31a-4cb1-9468-ffd82c8349ae:term_by_term","label":"(c) Term-by-Term Calculation","steps":[{"type":"text","order":0,"title":"Understand the summation task","content":"We need to find the value of the sum of the first four terms of our sequence, $\\sum_{r=1}^4 x_r$, in terms of $k$:\n\n$$\\sum_{r=1}^4 x_r = x_1 + x_2 + x_3 + x_4$$\n\nTo do this, we will find each individual term starting from $x_1$ up to $x_4$ using the recurrence relation, and then add them together.","highlights":[{"text":"Find $\\sum_{r=1}^4 x_r$ in terms of $k$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=x1-val}{\\color{@sky}{x_1 = k}}$"},{"latex":"$$\\htmlData{anchor=x2-val}{\\color{@amber}{x_2}} = 4(\\color{@sky}{x_1}) - 1 = \\htmlData{anchor=x2-expr}{\\color{@amber}{4k - 1}}$"},{"latex":"$$\\htmlData{anchor=x3-val}{\\color{@purple}{x_3}} = 4(\\color{@amber}{x_2}) - 1 = 4(\\color{@amber}{4k - 1}) - 1 = \\htmlData{anchor=x3-expr}{\\color{@purple}{16k - 5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x1-val"},{"color":"amber","style":"text-color","anchor":"x2-expr"},{"color":"purple","style":"text-color","anchor":"x3-expr"}],"connections":[{"to":"x2-val","from":"x1-val","color":"sky","label":"substitute x1","style":"solid"},{"to":"x3-val","from":"x2-expr","color":"amber","label":"substitute x2","style":"solid"}]},"order":1,"title":"Find the first three terms","description":"Using the relation $x_n = 4x_{n-1} - 1$, we can determine the first three terms step-by-step in terms of $k$."},{"type":"text","order":2,"title":"Calculate the fourth term","content":"Now we calculate the fourth term, $x_4$, by substituting our expression for $x_3 = 16k - 5$ into the recurrence relation:\n\n$$\\begin{aligned} x_4 &= 4x_3 - 1 \\\\ &= 4(16k - 5) - 1 \\\\ &= 64k - 20 - 1 \\\\ &= 64k - 21 \\end{aligned}$$\n\nThis calculation earns you the first method mark (**M1**)."},{"type":"text","order":3,"title":"Sum the four terms","content":"Now, we write down the sum of the first four terms:\n\n$$\\sum_{r=1}^4 x_r = x_1 + x_2 + x_3 + x_4$$\n\nSubstituting our expressions for each term, we get:\n\n$$\\sum_{r=1}^4 x_r = (k) + (4k - 1) + (16k - 5) + (64k - 21)$$\n\nWriting this sum earns the second method mark (**M1**)."},{"type":"text","order":4,"title":"Simplify the expression","content":"To get our final answer, we group the $k$ terms and the constant terms together:\n\n$$\\begin{aligned} \\sum_{r=1}^4 x_r &= (1 + 4 + 16 + 64)k + (-1 - 5 - 21) \\\\ &= 85k - 27 \\end{aligned}$$\n\nThis simplified final result earns the accuracy mark (**A1**)."}]},{"id":"087e4553-a31a-4cb1-9468-ffd82c8349ae:closed_form_gp","label":"(c) Finding the General Term","steps":[{"type":"text","order":0,"title":"Analyze the recurrence relation","content":"We are given the first-order linear recurrence relation:\n\n$$x_n = 4x_{n-1} - 1 \\quad \\text{for } n \\geq 2$$ \n\nwith initial term $x_1 = k$. To find the sum $\\sum_{r=1}^4 x_r$, we will first find a closed-form expression for the general term $x_n$ by representing it as a shifted geometric sequence."},{"type":"text","order":1,"title":"Find the constant shift","content":"Let's find a constant $c$ such that the sequence $y_n = x_n - c$ is geometric. If $y_n$ is geometric with a common ratio of $4$, then:\n\n$$y_n = 4y_{n-1}$$\n\nSubstituting $y_n = x_n - c$ into this relation gives:\n\n$$x_n - c = 4(x_{n-1} - c)$$\n\n$$x_n = 4x_{n-1} - 3c$$\n\nComparing this with our original recurrence relation $x_n = 4x_{n-1} - 1$, we get:\n\n$$-3c = -1 \\implies c = \\frac{1}{3}$$"},{"type":"text","order":2,"title":"Define the shifted sequence","content":"Using our value of $c = \\frac{1}{3}$, we can define the geometric sequence $y_n$ as:\n\n$$y_n = x_n - \\frac{1}{3}$$\n\nLet's find the first term $y_1$:\n\n$$y_1 = x_1 - \\frac{1}{3} = k - \\frac{1}{3}$$\n\nSince $y_n$ has a common ratio $r = 4$, the general term $y_n$ is:\n\n$$y_n = y_1 \\cdot r^{n-1} = \\left(k - \\frac{1}{3}\\right) 4^{n-1}$$"},{"type":"text","order":3,"title":"Express the general term","content":"Now we can write the general term $x_n$ in closed form by adding the shift back:\n\n$$x_n = y_n + \\frac{1}{3} = \\left(k - \\frac{1}{3}\\right) 4^{n-1} + \\frac{1}{3}$$\n\nLet's verify this formula for $n = 1$:\n\n$$x_1 = \\left(k - \\frac{1}{3}\\right) 4^0 + \\frac{1}{3} = k$$"},{"type":"text","order":4,"title":"Sum the sequence","content":"We need to find the sum of the first four terms, $\\sum_{r=1}^4 x_r$. Using our general term expression:\n\n$$\\sum_{r=1}^4 x_r = \\sum_{r=1}^4 \\left[ \\left(k - \\frac{1}{3}\\right) 4^{r-1} + \\frac{1}{3} \\right]$$\n\nWe can split this into two parts:\n\n$$\\sum_{r=1}^4 x_r = \\left(k - \\frac{1}{3}\\right) \\sum_{r=1}^4 4^{r-1} + \\sum_{r=1}^4 \\frac{1}{3}$$"},{"type":"text","order":5,"title":"Calculate the sums","content":"First, evaluate the geometric sum $\\sum_{r=1}^4 4^{r-1}$ using the geometric series sum formula $S_n = \\frac{a(q^n - 1)}{q - 1}$ where $a = 1, q = 4, n = 4$:\n\n$$\\sum_{r=1}^4 4^{r-1} = \\frac{1(4^4 - 1)}{4 - 1} = \\frac{256 - 1}{3} = \\frac{255}{3} = 85$$\n\nNext, sum the constant term $\\frac{1}{3}$ over $4$ terms:\n\n$$\\sum_{r=1}^4 \\frac{1}{3} = 4 \\times \\frac{1}{3} = \\frac{4}{3}$$"},{"type":"text","order":6,"title":"Combine and simplify","content":"Now substitute these values back into our sum expression:\n\n$$\\begin{aligned} \\sum_{r=1}^4 x_r &= 85\\left(k - \\frac{1}{3}\\right) + \\frac{4}{3} \\\\ &= 85k - \\frac{85}{3} + \\frac{4}{3} \\\\ &= 85k - \\frac{81}{3} \\\\ &= 85k - 27 \\end{aligned}$$\n\nThis matches the simplified sum of the individual terms: $k + (4k-1) + (16k-5) + (64k-21) = 85k-27$."}]}],"generatedAt":"2026-06-11T03:21:03.170Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":65},"walkthroughVideo":null},{"id":"02e9df4b-1c3b-4086-89c5-ff1e95c0f4d4","revisionId":"27885bd0-bab9-4e52-a70f-c73fe6496ecf","questionNumber":null,"questionType":"LA","specification":"The first three terms of an arithmetic series are $(2x-3)$, $(5x-2)$ and $(10x-7)$ respectively.","difficulty":1,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"15752458-3569-4fb8-b43a-fa7bb028b16d","label":"a","content":"Find the value of the constant $x$.","markscheme":"- Use common difference property: $(5x-2) - (2x-3) = (10x-7) - (5x-2)$ M1\n- $3x + 1 = 5x - 5 \\implies 2x = 6 \\implies x = 3$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Sets up an equation using the common difference property $(u_2 - u_1 = u_3 - u_2)$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly solves the linear equation to find $x = 3$.","code":"A"}],"finalAnswers":["3"],"generatedAt":"2026-05-06T14:05:20.742Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"common_difference","label":"Equating Common Differences","steps":[{"type":"text","order":0,"title":"Understand common differences","content":"For any three consecutive terms in an arithmetic sequence, $T_1$, $T_2$, and $T_3$, the difference between successive terms is constant. This is called the common difference, $d$.\n\nTherefore, we can set up an equation by equating the difference between the second and first terms to the difference between the third and second terms:\n\n$$T_2 - T_1 = T_3 - T_2$$","highlights":[{"text":"$$(2x-3)$","location":"specification"},{"text":"$$(5x-2)$","location":"specification"},{"text":"$$(10x-7)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$T_1 = \\htmlData{anchor=t1}{\\color{@sky}{2x - 3}}, \\quad T_2 = \\htmlData{anchor=t2}{\\color{@amber}{5x - 2}}, \\quad T_3 = \\htmlData{anchor=t3}{\\color{@purple}{10x - 7}}$"},{"latex":"$$(\\htmlData{anchor=t2-sub}{\\color{@amber}{5x - 2}}) - (\\htmlData{anchor=t1-sub}{\\color{@sky}{2x - 3}}) = (\\htmlData{anchor=t3-sub}{\\color{@purple}{10x - 7}}) - (\\htmlData{anchor=t2-sub2}{\\color{@amber}{5x - 2}})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t1"},{"color":"amber","style":"text-color","anchor":"t2"},{"color":"purple","style":"text-color","anchor":"t3"}],"connections":[{"to":"t1-sub","from":"t1","color":"sky","label":"T₁","style":"solid"},{"to":"t2-sub","from":"t2","color":"amber","label":"T₂","style":"solid"},{"to":"t3-sub","from":"t3","color":"purple","label":"T₃","style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute our expressions for $T_1$, $T_2$, and $T_3$ into the common difference formula to write an equation in terms of $x$."},{"type":"text","order":2,"title":"Simplify both sides","content":"Now we simplify each side of the equation carefully by expanding and collecting like terms:\n\n$$5x - 2 - 2x + 3 = 10x - 7 - 5x + 2$$\n\n$$3x + 1 = 5x - 5$$\n\nThis step earns the method mark (**M1**) for correctly setting up and expanding the common difference equation."},{"type":"text","order":3,"title":"Solve for x","content":"Rearrange the simplified equation to solve for the constant $x$:\n\n$$1 + 5 = 5x - 3x$$\n\n$$6 = 2x$$\n\n$$x = 3$$\n\nThis final answer earns the accuracy mark (**A1**)."},{"type":"text","order":4,"title":"Find the key values","content":"With $x = 3$, we find the first term, $a$, and the common difference, $d$, of our arithmetic series:\n\n$$a = 2(3) - 3 = 3$$\n\n$$d = (5(3) - 2) - 3 = 10$$\n\nThis step earns the first mark (**M1**) for Part 2."},{"type":"text","order":5,"title":"Calculate the 25-term sum","content":"We use the sum of an arithmetic series formula from our formula booklet:\n\n$$S_n = \\frac{n}{2}[2a + (n - 1)d]$$\n\nSubstituting $n = 25$, $a = 3$, and $d = 10$:\n\n$$S_{25} = \\frac{25}{2}[2(3) + 24(10)]$$\n\n$$S_{25} = 12.5 \\times 246 = 3075$$\n\nThis completes Part 2, earning the final two marks (**M1** and **A1**)."}]},{"id":"arithmetic_mean","label":"Using Arithmetic Mean Property","steps":[{"type":"text","order":0,"title":"Recall arithmetic mean property","content":"For any three consecutive terms of an arithmetic progression, say $a$, $b$, and $c$, the middle term is the arithmetic mean of the other two terms. This relation is written as:\n\n$$b = \\frac{a + c}{2} \\quad \\text{or} \\quad 2b = a + c$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$a = \\htmlData{anchor=term1}{\\color{@sky}{2x - 3}}, \\quad b = \\htmlData{anchor=term2}{\\color{@amber}{5x - 2}}, \\quad c = \\htmlData{anchor=term3}{\\color{@purple}{10x - 7}}$"},{"latex":"$$2(\\htmlData{anchor=term2-sub}{\\color{@amber}{5x - 2}}) = (\\htmlData{anchor=term1-sub}{\\color{@sky}{2x - 3}}) + (\\htmlData{anchor=term3-sub}{\\color{@purple}{10x - 7}})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"term1"},{"color":"amber","style":"text-color","anchor":"term2"},{"color":"purple","style":"text-color","anchor":"term3"}],"connections":[{"to":"term1-sub","from":"term1","color":"sky","label":null,"style":"solid"},{"to":"term2-sub","from":"term2","color":"amber","label":null,"style":"solid"},{"to":"term3-sub","from":"term3","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute the three consecutive terms into the arithmetic mean relationship $2b = a + c$."},{"type":"text","order":2,"title":"Simplify and solve","content":"We expand the left-hand side and collect like terms on the right-hand side:\n\n$$10x - 4 = 12x - 10$$\n\nNow, rearrange the terms to solve for $x$:\n\n$$-2x = -6$$\n\n$$x = 3$$\n\nThis calculation secures the final **A1** mark for Part 1."},{"type":"text","order":3,"title":"Find the key values","content":"With $x = 3$, we determine the first term, $a$, and common difference, $d$:\n\n$$a = 2(3) - 3 = 3$$\n\n$$d = (5(3) - 2) - 3 = 10$$\n\nThis step secures the first method mark (**M1**) for Part 2."},{"type":"text","order":4,"title":"Calculate the 25-term sum","content":"Using the arithmetic progression sum formula:\n\n$$S_n = \\frac{n}{2}[2a + (n-1)d]$$\n\nSubstituting $n = 25$, $a = 3$, and $d = 10$:\n\n$$S_{25} = \\frac{25}{2}[2(3) + 24(10)] = 3075$$\n\nThis completes the calculation, securing the remaining marks (**M1** and **A1**)."}]}],"generatedAt":"2026-06-09T02:59:34.207Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"ac4ab19d-c96d-45f7-9608-a341d0f8257a","label":"b","content":"Find the sum of the first 25 terms of the series.","markscheme":"- Find $a = 2(3)-3 = 3$ and $d = (5(3)-2) - 3 = 10$ M1\n- $S_{25} = \\frac{25}{2}(2(3) + 24(10))$ M1\n- $S_{25} = 12.5(6 + 240) = 3075$ A1","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p3","marks":1,"description":"Finds the first term $a$ and the common difference $d$ using their value of $x$.","code":"M"},{"id":"p4","marks":1,"description":"Correctly substitutes $a$, $d$, and $n=25$ into the arithmetic sum formula.","code":"M"},{"id":"p5","marks":1,"description":"Calculates the correct final sum of 3075.","code":"A"}],"finalAnswers":["3075"],"generatedAt":"2026-05-06T14:05:20.742Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"standard_sum_formula","label":"Using Standard Sum Formula","steps":[{"type":"text","order":0,"title":"Solve for x","content":"To find the first term and the common difference, we first need to determine the value of $x$. Since the terms $(2x-3)$, $(5x-2)$, and $(10x-7)$ form an arithmetic series, the difference between consecutive terms is constant:\n\n$$(5x - 2) - (2x - 3) = (10x - 7) - (5x - 2)$$\n\nSimplifying both sides:\n\n$$3x + 1 = 5x - 5$$\n\nRearranging to solve for $x$:\n\n$$2x = 6 \\implies x = 3$$","highlights":[{"text":"(2x-3)","location":"specification"},{"text":"(5x-2)","location":"specification"},{"text":"(10x-7)","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\text{With } \\htmlData{anchor=x-val}{x = 3}:"},{"latex":"\\text{First term: } \\htmlData{anchor=a-def}{a} = 2(\\htmlData{anchor=x-sub-a}{3}) - 3 = \\htmlData{anchor=a-val}{\\color{@sky}{3}}"},{"latex":"\\text{Second term: } u_2 = 5(3) - 2 = 13"},{"latex":"\\text{Common difference: } \\htmlData{anchor=d-def}{d} = 13 - \\htmlData{anchor=a-sub-d}{\\color{@sky}{3}} = \\htmlData{anchor=d-val}{\\color{@amber}{10}}"}],"highlights":[{"color":"sky","style":"box","anchor":"a-val"},{"color":"amber","style":"box","anchor":"d-val"}],"connections":[{"to":"x-sub-a","from":"x-val","color":"sky","label":"Substitute x","style":"dashed"},{"to":"a-sub-d","from":"a-val","color":"sky","label":"First term","style":"solid"}]},"order":1,"title":"Find the first term and common difference","description":"Substitute $x = 3$ back into our expressions to find the first term $a$ and the common difference $d$. You earn the first mark (M1) for finding both values correctly."},{"type":"text","order":2,"title":"Recall the arithmetic sum formula","content":"From your formula booklet, the sum of the first $n$ terms of an arithmetic series is given by:\n\n$$S_n = \\frac{n}{2}\\left(2a + (n-1)d\\right)$$\n\nWe need to find the sum of the first $25$ terms, so we will use:\n* $n = 25$\n* $a = 3$\n* $d = 10$","highlights":[{"text":"first 25 terms","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$S_{25} = \\frac{\\htmlData{anchor=n-val}{25}}{2}\\left(2(\\htmlData{anchor=a-sub}{\\color{@sky}{3}}) + (\\htmlData{anchor=n-sub}{25} - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{10}}\\right)$"},{"latex":"$$S_{25} = 12.5 \\times \\left(6 + 24 \\times 10\\right)$"},{"latex":"$$S_{25} = 12.5 \\times 246 = \\htmlData{anchor=final-ans}{3075}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-sub"},{"color":"amber","style":"text-color","anchor":"d-sub"},{"color":"green","style":"box","anchor":"final-ans"}],"connections":[{"to":"n-sub","from":"n-val","color":"purple","label":"n = 25","style":"dashed"}]},"order":3,"title":"Substitute and calculate","description":"Substitute the parameters into the formula (M1) and simplify to find the final sum of 3075 (A1)."}]},{"id":"last_term_formula","label":"Using First and Last Terms","steps":[{"type":"text","order":0,"title":"Find the first term","content":"From the first part of the question, we know that $x = 3$.\n\nTo find the first term, $a$, of the arithmetic series, we substitute $x = 3$ into the expression for the first term, $(2x - 3)$:\n\n$$a = 2(3) - 3 = 3$$"},{"type":"text","order":1,"title":"Find the common difference","content":"Next, we find the common difference, $d$. First, calculate the second term by substituting $x = 3$ into $(5x - 2)$:\n\n$$\\text{Second term} = 5(3) - 2 = 13$$\n\nNow, find the difference between the second term and the first term ($a = 3$):\n\n$$d = 13 - 3 = 10$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-val}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-val}{\\color{@amber}{d}}$"},{"latex":"$$u_{25} = \\htmlData{anchor=a-sub}{\\color{@sky}{3}} + (25 - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{10}}$"},{"latex":"$$u_{25} = 3 + 24(10) = 243$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"amber","style":"text-color","anchor":"d-val"}],"connections":[{"to":"a-sub","from":"a-val","color":"sky","label":null,"style":"solid"},{"to":"d-sub","from":"d-val","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Calculate the 25th term","description":"We can find the 25th term, $u_{25}$, using the $n$-th term formula $u_n = a + (n-1)d$ with $n = 25$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$S_n = \\frac{n}{2}(\\htmlData{anchor=a-sum-val}{\\color{@sky}{a}} + \\htmlData{anchor=last-val}{\\color{@purple}{u_n}})$"},{"latex":"$$S_{25} = \\frac{25}{2}(\\htmlData{anchor=a-sum-sub}{\\color{@sky}{3}} + \\htmlData{anchor=last-sub}{\\color{@purple}{243}})$"},{"latex":"$$S_{25} = 12.5 \\times 246 = 3075$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-sum-val"},{"color":"purple","style":"text-color","anchor":"last-val"}],"connections":[{"to":"a-sum-sub","from":"a-sum-val","color":"sky","label":null,"style":"solid"},{"to":"last-sub","from":"last-val","color":"purple","label":null,"style":"solid"}]},"order":3,"title":"Calculate the sum","description":"Now we use the alternative arithmetic series sum formula $S_n = \\frac{n}{2}(a + u_n)$, where $a$ is the first term and $u_n$ is the last (25th) term."}]}],"generatedAt":"2026-06-09T03:00:53.705Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"15752458-3569-4fb8-b43a-fa7bb028b16d:common_difference","label":"(a) Equating Common Differences","steps":[{"type":"text","order":0,"title":"Understand common differences","content":"For any three consecutive terms in an arithmetic sequence, $T_1$, $T_2$, and $T_3$, the difference between successive terms is constant. This is called the common difference, $d$.\n\nTherefore, we can set up an equation by equating the difference between the second and first terms to the difference between the third and second terms:\n\n$$T_2 - T_1 = T_3 - T_2$$","highlights":[{"text":"$$(2x-3)$","location":"specification"},{"text":"$$(5x-2)$","location":"specification"},{"text":"$$(10x-7)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$T_1 = \\htmlData{anchor=t1}{\\color{@sky}{2x - 3}}, \\quad T_2 = \\htmlData{anchor=t2}{\\color{@amber}{5x - 2}}, \\quad T_3 = \\htmlData{anchor=t3}{\\color{@purple}{10x - 7}}$"},{"latex":"$$(\\htmlData{anchor=t2-sub}{\\color{@amber}{5x - 2}}) - (\\htmlData{anchor=t1-sub}{\\color{@sky}{2x - 3}}) = (\\htmlData{anchor=t3-sub}{\\color{@purple}{10x - 7}}) - (\\htmlData{anchor=t2-sub2}{\\color{@amber}{5x - 2}})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t1"},{"color":"amber","style":"text-color","anchor":"t2"},{"color":"purple","style":"text-color","anchor":"t3"}],"connections":[{"to":"t1-sub","from":"t1","color":"sky","label":"T₁","style":"solid"},{"to":"t2-sub","from":"t2","color":"amber","label":"T₂","style":"solid"},{"to":"t3-sub","from":"t3","color":"purple","label":"T₃","style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute our expressions for $T_1$, $T_2$, and $T_3$ into the common difference formula to write an equation in terms of $x$."},{"type":"text","order":2,"title":"Simplify both sides","content":"Now we simplify each side of the equation carefully by expanding and collecting like terms:\n\n$$5x - 2 - 2x + 3 = 10x - 7 - 5x + 2$$\n\n$$3x + 1 = 5x - 5$$\n\nThis step earns the method mark (**M1**) for correctly setting up and expanding the common difference equation."},{"type":"text","order":3,"title":"Solve for x","content":"Rearrange the simplified equation to solve for the constant $x$:\n\n$$1 + 5 = 5x - 3x$$\n\n$$6 = 2x$$\n\n$$x = 3$$\n\nThis final answer earns the accuracy mark (**A1**)."},{"type":"text","order":4,"title":"Find the key values","content":"With $x = 3$, we find the first term, $a$, and the common difference, $d$, of our arithmetic series:\n\n$$a = 2(3) - 3 = 3$$\n\n$$d = (5(3) - 2) - 3 = 10$$\n\nThis step earns the first mark (**M1**) for Part 2."},{"type":"text","order":5,"title":"Calculate the 25-term sum","content":"We use the sum of an arithmetic series formula from our formula booklet:\n\n$$S_n = \\frac{n}{2}[2a + (n - 1)d]$$\n\nSubstituting $n = 25$, $a = 3$, and $d = 10$:\n\n$$S_{25} = \\frac{25}{2}[2(3) + 24(10)]$$\n\n$$S_{25} = 12.5 \\times 246 = 3075$$\n\nThis completes Part 2, earning the final two marks (**M1** and **A1**)."}]},{"id":"15752458-3569-4fb8-b43a-fa7bb028b16d:arithmetic_mean","label":"(a) Using Arithmetic Mean Property","steps":[{"type":"text","order":0,"title":"Recall arithmetic mean property","content":"For any three consecutive terms of an arithmetic progression, say $a$, $b$, and $c$, the middle term is the arithmetic mean of the other two terms. This relation is written as:\n\n$$b = \\frac{a + c}{2} \\quad \\text{or} \\quad 2b = a + c$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$a = \\htmlData{anchor=term1}{\\color{@sky}{2x - 3}}, \\quad b = \\htmlData{anchor=term2}{\\color{@amber}{5x - 2}}, \\quad c = \\htmlData{anchor=term3}{\\color{@purple}{10x - 7}}$"},{"latex":"$$2(\\htmlData{anchor=term2-sub}{\\color{@amber}{5x - 2}}) = (\\htmlData{anchor=term1-sub}{\\color{@sky}{2x - 3}}) + (\\htmlData{anchor=term3-sub}{\\color{@purple}{10x - 7}})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"term1"},{"color":"amber","style":"text-color","anchor":"term2"},{"color":"purple","style":"text-color","anchor":"term3"}],"connections":[{"to":"term1-sub","from":"term1","color":"sky","label":null,"style":"solid"},{"to":"term2-sub","from":"term2","color":"amber","label":null,"style":"solid"},{"to":"term3-sub","from":"term3","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute the three consecutive terms into the arithmetic mean relationship $2b = a + c$."},{"type":"text","order":2,"title":"Simplify and solve","content":"We expand the left-hand side and collect like terms on the right-hand side:\n\n$$10x - 4 = 12x - 10$$\n\nNow, rearrange the terms to solve for $x$:\n\n$$-2x = -6$$\n\n$$x = 3$$\n\nThis calculation secures the final **A1** mark for Part 1."},{"type":"text","order":3,"title":"Find the key values","content":"With $x = 3$, we determine the first term, $a$, and common difference, $d$:\n\n$$a = 2(3) - 3 = 3$$\n\n$$d = (5(3) - 2) - 3 = 10$$\n\nThis step secures the first method mark (**M1**) for Part 2."},{"type":"text","order":4,"title":"Calculate the 25-term sum","content":"Using the arithmetic progression sum formula:\n\n$$S_n = \\frac{n}{2}[2a + (n-1)d]$$\n\nSubstituting $n = 25$, $a = 3$, and $d = 10$:\n\n$$S_{25} = \\frac{25}{2}[2(3) + 24(10)] = 3075$$\n\nThis completes the calculation, securing the remaining marks (**M1** and **A1**)."}]},{"id":"ac4ab19d-c96d-45f7-9608-a341d0f8257a:standard_sum_formula","label":"(b) Using Standard Sum Formula","steps":[{"type":"text","order":0,"title":"Solve for x","content":"To find the first term and the common difference, we first need to determine the value of $x$. Since the terms $(2x-3)$, $(5x-2)$, and $(10x-7)$ form an arithmetic series, the difference between consecutive terms is constant:\n\n$$(5x - 2) - (2x - 3) = (10x - 7) - (5x - 2)$$\n\nSimplifying both sides:\n\n$$3x + 1 = 5x - 5$$\n\nRearranging to solve for $x$:\n\n$$2x = 6 \\implies x = 3$$","highlights":[{"text":"(2x-3)","location":"specification"},{"text":"(5x-2)","location":"specification"},{"text":"(10x-7)","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\text{With } \\htmlData{anchor=x-val}{x = 3}:"},{"latex":"\\text{First term: } \\htmlData{anchor=a-def}{a} = 2(\\htmlData{anchor=x-sub-a}{3}) - 3 = \\htmlData{anchor=a-val}{\\color{@sky}{3}}"},{"latex":"\\text{Second term: } u_2 = 5(3) - 2 = 13"},{"latex":"\\text{Common difference: } \\htmlData{anchor=d-def}{d} = 13 - \\htmlData{anchor=a-sub-d}{\\color{@sky}{3}} = \\htmlData{anchor=d-val}{\\color{@amber}{10}}"}],"highlights":[{"color":"sky","style":"box","anchor":"a-val"},{"color":"amber","style":"box","anchor":"d-val"}],"connections":[{"to":"x-sub-a","from":"x-val","color":"sky","label":"Substitute x","style":"dashed"},{"to":"a-sub-d","from":"a-val","color":"sky","label":"First term","style":"solid"}]},"order":1,"title":"Find the first term and common difference","description":"Substitute $x = 3$ back into our expressions to find the first term $a$ and the common difference $d$. You earn the first mark (M1) for finding both values correctly."},{"type":"text","order":2,"title":"Recall the arithmetic sum formula","content":"From your formula booklet, the sum of the first $n$ terms of an arithmetic series is given by:\n\n$$S_n = \\frac{n}{2}\\left(2a + (n-1)d\\right)$$\n\nWe need to find the sum of the first $25$ terms, so we will use:\n* $n = 25$\n* $a = 3$\n* $d = 10$","highlights":[{"text":"first 25 terms","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$S_{25} = \\frac{\\htmlData{anchor=n-val}{25}}{2}\\left(2(\\htmlData{anchor=a-sub}{\\color{@sky}{3}}) + (\\htmlData{anchor=n-sub}{25} - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{10}}\\right)$"},{"latex":"$$S_{25} = 12.5 \\times \\left(6 + 24 \\times 10\\right)$"},{"latex":"$$S_{25} = 12.5 \\times 246 = \\htmlData{anchor=final-ans}{3075}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-sub"},{"color":"amber","style":"text-color","anchor":"d-sub"},{"color":"green","style":"box","anchor":"final-ans"}],"connections":[{"to":"n-sub","from":"n-val","color":"purple","label":"n = 25","style":"dashed"}]},"order":3,"title":"Substitute and calculate","description":"Substitute the parameters into the formula (M1) and simplify to find the final sum of 3075 (A1)."}]},{"id":"ac4ab19d-c96d-45f7-9608-a341d0f8257a:last_term_formula","label":"(b) Using First and Last Terms","steps":[{"type":"text","order":0,"title":"Find the first term","content":"From the first part of the question, we know that $x = 3$.\n\nTo find the first term, $a$, of the arithmetic series, we substitute $x = 3$ into the expression for the first term, $(2x - 3)$:\n\n$$a = 2(3) - 3 = 3$$"},{"type":"text","order":1,"title":"Find the common difference","content":"Next, we find the common difference, $d$. First, calculate the second term by substituting $x = 3$ into $(5x - 2)$:\n\n$$\\text{Second term} = 5(3) - 2 = 13$$\n\nNow, find the difference between the second term and the first term ($a = 3$):\n\n$$d = 13 - 3 = 10$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-val}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-val}{\\color{@amber}{d}}$"},{"latex":"$$u_{25} = \\htmlData{anchor=a-sub}{\\color{@sky}{3}} + (25 - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{10}}$"},{"latex":"$$u_{25} = 3 + 24(10) = 243$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"amber","style":"text-color","anchor":"d-val"}],"connections":[{"to":"a-sub","from":"a-val","color":"sky","label":null,"style":"solid"},{"to":"d-sub","from":"d-val","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Calculate the 25th term","description":"We can find the 25th term, $u_{25}$, using the $n$-th term formula $u_n = a + (n-1)d$ with $n = 25$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$S_n = \\frac{n}{2}(\\htmlData{anchor=a-sum-val}{\\color{@sky}{a}} + \\htmlData{anchor=last-val}{\\color{@purple}{u_n}})$"},{"latex":"$$S_{25} = \\frac{25}{2}(\\htmlData{anchor=a-sum-sub}{\\color{@sky}{3}} + \\htmlData{anchor=last-sub}{\\color{@purple}{243}})$"},{"latex":"$$S_{25} = 12.5 \\times 246 = 3075$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-sum-val"},{"color":"purple","style":"text-color","anchor":"last-val"}],"connections":[{"to":"a-sum-sub","from":"a-sum-val","color":"sky","label":null,"style":"solid"},{"to":"last-sub","from":"last-val","color":"purple","label":null,"style":"solid"}]},"order":3,"title":"Calculate the sum","description":"Now we use the alternative arithmetic series sum formula $S_n = \\frac{n}{2}(a + u_n)$, where $a$ is the first term and $u_n$ is the last (25th) 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Sequences and Series

Question 20

Sequences and Series Questions

Sequences and Series

Question 20

Sequences and Series Questions