A sequence is a list of numbers that follow a pattern. Different types of sequences follow different patterns, for example some increase by a constant amount term to term and others follow a rule of multiplying the previous term by a constant find the next term.
", "rulesets": {}, "variable_groups": [{"variables": ["a1", "b1", "c2"], "name": "seq' eqn' question c)"}], "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b"], "parts": [{"variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "minAnswers": 0, "shuffleChoices": false, "answers": ["Quadratic
", "Linear
"], "prompt": "For each of these sequences state whether they are linear or quadratic.
", "layout": {"type": "all", "expression": ""}, "maxAnswers": 0, "showCorrectAnswer": true, "minMarks": 0, "displayType": "radiogroup", "warningType": "none", "choices": ["ii) $\\var{m[3]*5}, \\var{m[3]*6}, \\var{m[3]*7}, \\var{m[3]*8}...$
", "iii) $\\var{m[2]*10}, \\var{m[2]*11}, \\var{m[2]*12}...$
", "iv) $\\var{b[4]*1^2+n[4]*1+c[4]}, \\var{b[4]*2^2+n[4]*2+c[4]}, \\var{b[4]*3^2+n[4]*3+c[4]}, \\var{b[4]*4^2+n[4]*4+c[4]}...$
"], "type": "m_n_x", "matrix": [["1", 0], [0, "1"], [0, "1"]], "showFeedbackIcon": true, "shuffleAnswers": false, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "minAnswers": 0, "shuffleChoices": false, "answers": ["Arithmetic
", "Geometric
"], "prompt": "Which of these sequences are arithmetic and which are geometric?
", "layout": {"type": "all", "expression": ""}, "maxAnswers": 0, "showCorrectAnswer": true, "minMarks": 0, "displayType": "radiogroup", "warningType": "none", "choices": ["i) $\\var{c[0]}, \\var{c[0]^2}, \\var{c[0]^3}, \\var{c[0]^4}\\ldots$
", "ii) $\\var{c[2]*c[3]}, \\var{c[2]*c[3]^2}, \\var{c[2]*c[3]^3}, \\var{c[2]*c[3]^4}\\ldots$
", "iii) $\\var{c[1]*5}, \\var{c[1]*6}, \\var{c[1]*7}, \\var{c[1]*8}, \\ldots$
", "iv) $\\var{c[2]*8}, \\var{c[2]*9}, \\var{c[2]*10}, \\var{c[2]*11} \\ldots$
"], "type": "m_n_x", "matrix": [[0, "1"], [0, "1"], ["1", 0], ["1", 0]], "showFeedbackIcon": true, "shuffleAnswers": false, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "stepsPenalty": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "15", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "15", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "21", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "21", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "28", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "28", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}], "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "type": "information", "prompt": "The $n^{th}$ term of the triangle sequence is
\n\\[\\frac{n(n+1)}{2}\\text{.}\\]
", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}], "prompt": "Write out the next three terms in the triangle sequence
\n$1, 3, 6, 10,$ [[0]], [[1]], [[2]]
", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill"}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "Find the $\\var{ci[0]}$th term of the sequence with formula:
\n\\[a_n=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\]
\n$a_{\\var{ci[0]}}=$ [[0]]
", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "showpreview": true, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "answer": "{a1}{ci[0]}({ci[0]}+{b1})/{c2}", "showCorrectAnswer": true, "expectedvariablenames": [], "checkvariablenames": false, "marks": "1", "variableReplacements": [], "type": "jme", "vsetrange": [0, 1], "showFeedbackIcon": true, "vsetrangepoints": 5, "scripts": {}}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "Write out the next three terms of the following sequences.
\ni)
\n$1, 4, 9, 16, 25,$ [[0]], [[1]], [[2]]
\nii)
\n$1, 8, 27, 64,$ [[3]], [[4]], [[5]]
", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "36", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "36", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "49", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "49", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "64", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "64", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "125", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "125", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "216", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "216", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "minValue": "343", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "343", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "allowFractions": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}], "functions": {}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Differentiate between linear and quadratic sequences and arithmetic and geometric sequences through a series of multiple choice questions. Spot different patterns in sequences like the triangle sequence, square sequence and cubic sequence and then use this pattern to find the next three terms in each of the sequences.
"}, "type": "question", "variables": {"b1": {"group": "seq' eqn' question c)", "description": "", "definition": "random(2..7)c2", "name": "b1", "templateType": "anything"}, "ni": {"group": "Ungrouped variables", "description": "", "definition": "repeat(random(19..40),10)", "name": "ni", "templateType": "anything"}, "c2": {"group": "seq' eqn' question c)", "description": "", "definition": "random(2..5)", "name": "c2", "templateType": "anything"}, "n": {"group": "Ungrouped variables", "description": "", "definition": "repeat(random(1..4),7)", "name": "n", "templateType": "anything"}, "c": {"group": "Ungrouped variables", "description": "", "definition": "repeat(random(3..13 except[10]),8)", "name": "c", "templateType": "anything"}, "m": {"group": "Ungrouped variables", "description": "", "definition": "repeat(random(2..10),5)", "name": "m", "templateType": "anything"}, "a1": {"group": "seq' eqn' question c)", "description": "", "definition": "random(2..5)", "name": "a1", "templateType": "anything"}, "ci": {"group": "Ungrouped variables", "description": "", "definition": "repeat(random(3..9 except 5)c2,10)", "name": "ci", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "repeat(random(2..4), 5)", "name": "b", "templateType": "anything"}}, "advice": "A linear sequence is a sequence with a constant common difference whilst a quadratic sequence has a difference which increases or decreases by a constant value. We can calculate the differences between terms by constructing tables. These tables can then be used to decide whether the sequence is linear or quadratic.
\ni)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\var{b[2]+n[2]+c[2]}$ | \n$\\var{b[2]*2^2+n[2]*2+c[2]}$ | \n$\\var{b[2]*3^2+n[2]*3+c[2]}$ | \n$\\var{b[2]*4^2+n[2]*4+c[2]}$ | \n
| Differences between terms | \n\n | $\\var{b[2]*2^2+n[2]*2+c[2]-b[2]*1^2-n[2]*1-c[2]}$ | \n$\\var{b[2]*3^2+n[2]*3+c[2]-b[2]*2^2-n[2]*2-c[2]}$ | \n$\\var{b[2]*4^2+n[2]*4+c[2]-b[2]*3^2-n[2]*3-c[2]}$ | \n
If we look at the differences from this pattern we see they increase by $\\var{2*b[2]}$. As the original differences in the sequence increase by a constant the sequence is quadratic.
\n\nii)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\var{m[3]*5}$ | \n$\\var{m[3]*6}$ | \n$\\var{m[3]*7}$ | \n$\\var{m[3]*8}$ | \n
| Differences between terms | \n\n | $\\var{m[3]}$ | \n$\\var{m[3]}$ | \n$\\var{m[3]}$ | \n
This sequence increase by $\\var{m[3]}$ therefore is linear.
iii)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n
| $a_n$ | \n$\\var{m[2]*10}$ | \n$\\var{m[2]*11}$ | \n$\\var{m[2]*12}$ | \n
| Differences between terms | \n\n | $\\var{m[2]}$ | \n$\\var{m[2]}$ | \n
iv)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\var{b[4]*1^2+n[4]*1+c[4]}$ | \n$\\var{b[4]*2^2+n[4]*2+c[4]}$ | \n$\\var{b[4]*3^2+n[4]*3+c[4]}$ | \n$\\var{b[4]*4^2+n[4]*4+c[4]}$ | \n
| Differences between terms | \n\n | $\\var{3*b[4]+n[4]}$ | \n$\\var{5*b[4]+n[4]}$ | \n$\\var{7*b[4]+n[4]}$ | \n
If we look at the differences from this pattern we see they increase by $\\var{2*b[4]}$. As the original differences in the sequence increase by a constant the sequence is quadratic.
\nArithmetic sequences can be identified by the fact that they have a common difference, whereas geometric sequences have a common ratio.
\ni)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\var{c[0]}$ | \n$\\var{c[0]^2}$ | \n$ \\var{c[0]^3}$ | \n$ \\var{c[0]^4}$ | \n
| Differences between terms | \n\n | $\\var{c[0]^2-c[0]}$ | \n$\\var{c[0]^3-c[0]^2}$ | \n$\\var{c[0]^4-c[0]^3}$ | \n
This sequence has a common ratio of $\\var{c[0]}$ therefore the sequence is geometric.
\nii)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\var{c[2]*c[3]}$ | \n$\\var{c[2]*c[3]^2}$ | \n$ \\var{c[2]*c[3]^3}$ | \n$ \\var{c[2]*c[3]^4}$ | \n
| Differences between terms | \n\n | $\\var{c[2]*c[3]^2-c[2]*c[3]}$ | \n$\\var{c[2]*c[3]^3-c[2]*c[3]^2}$ | \n$\\var{c[2]*c[3]^4-c[2]*c[3]^3}$ | \n
This sequence has a common ratio of $\\var{c[3]}$ therefore the sequence is geometric.
\niii)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\\\var{c[1]*5}$ | \n$\\var{c[1]*6}$ | \n$ \\var{c[1]*7}$ | \n$ \\var{c[1]*8}$ | \n
| Differences between terms | \n\n | $\\var{c[1]}$ | \n$\\var{c[1]}$ | \n$\\var{c[1]}$ | \n
Each term in this sequence has a common difference of $\\var{c[1]}$ therefore the sequence is arithmetic.
\niv)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$\\var{c[2]*8}$ | \n$\\var{c[2]*9}$ | \n$ \\var{c[2]*10}$ | \n$ \\var{c[2]*11}$ | \n
| Differences between terms | \n\n | $\\var{c[2]}$ | \n$\\var{c[2]}$ | \n$\\var{c[2]}$ | \n
Each term in this sequence has a common difference of $\\var{c[2]}$ therefore the sequence is arithmetic.
\n\nWe can use a table to identify the triangle sequence by it's charecteristic of the difference between terms increasing by $1$ with each term.
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n$5$ | \n$6$ | \n$7$ | \n
| $a_n$ | \n$1$ | \n$3$ | \n$6$ | \n$10$ | \n$a_5$ | \n$a_6$ | \n$a_7$ | \n
| Differences between terms | \n\n | $2$ | \n$3$ | \n$4$ | \n$5$ | \n$6$ | \n$7$ | \n
We can then use this pattern to continue the sequence and identify the next three terms ($a_5$, $a_6$ and $a_7$).
\n\\[a_5=a_4+5=15\\]
\n\\[a_6=a_5+6=21\\]
\n\\[a_7=a_6+7=28\\]
\nOR
\nWe can use the formula for the $n^{th}$ term of the triangle sequence
\n\\[\\frac{n(n+1)}{2}\\text{.}\\]
\nto find the $5th, 6th\\; \\text{and}\\; 7th$ terms.
\n\\[\\begin{align}
\\frac{5(5+1)}{2}&=15\\\\
\\frac{6(6+1)}{2}&=21\\\\
\\frac{7(7+1)}{2}&=28
\\end{align}\\]
To find the answer to this question we must substitute our value for n ($\\var{ci[0]}$) into the formula for the sequence we have already been given:
\n\\[a_n=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\text{.}\\]
\nTherefore:
\n\\[
\\begin{align}
a_{\\var{ci[0]}}&=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\\\
&=\\frac{\\var{a1}\\times\\var{ci[0]}(\\var{ci[0]}+\\var{b1})}{\\var{c2}}\\\\
&=\\simplify{{{a1}*{ci[0]}*({ci[0]}+{b1})}/{c2}}
\\end{align}
\\]
\n
i)
\nWe can analyse the sequence by using a table to visualise each term with the difference between itself and the previous term.
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n$5$ | \n
| $a_n$ | \n$1$ | \n$4$ | \n$9$ | \n$16$ | \n$25$ | \n
| Differences between terms | \n\n | $3$ | \n$5$ | \n$7$ | \n$9$ | \n
The we observe that the difference between terms increases with each term, and by carefully observing the sequence, we may realise that this is a square sequence meaning each term has been squared to obtain its value.
\n\\[\\begin{align}
1^2&=1\\\\
2^2&=4\\\\
3^2&=9\\\\
4^2&=16\\\\
5^2&=25
\\end{align}\\]
Therefore to obtain the next three values we have to square the values
\n\\[\\begin{align}
6^2&=36\\\\
7^2&=49\\\\
8^2&=64\\text{.}
\\end{align}\\]
ii)
\n| $n$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n
| $a_n$ | \n$1$ | \n$8$ | \n$27$ | \n$64$ | \n
| Differences between terms | \n\n | $7$ | \n$21$ | \n$37$ | \n
The we observe that the difference between terms increases with each term, and by carefully observing the sequence, we may realise that this is a cubic sequence meaning each term has been cubed to obtain its value.
\n\\[\\begin{align}
1^3&=1\\\\
2^3&=8\\\\
3^3&=27\\\\
4^3&=64
\\end{align}\\]
Therefore to obtain the next three values we have to cube the values
\n\\[\\begin{align}
5^3&=125\\\\
6^3&=216\\\\
7^3&=343\\text{.}
\\end{align}\\]