// Numbas version: exam_results_page_options {"showQuestionGroupNames": false, "duration": 0, "timing": {"allowPause": true, "timeout": {"message": "", "action": "none"}, "timedwarning": {"message": "", "action": "none"}}, "type": "exam", "name": "Angharad's copy of Series quiz", "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-ordered", "questions": [{"name": "Arithmetic progression: The nth term of a series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Find the nth term of an Arithmetic progression

"}, "advice": "

If the difference between successive pairs of terms is a constant then the series under examination is an arithmetic progression.

\n

Ths first term is \$$a\$$ and the common difference is \$$d\$$.

\n

The formula for the nth term of the series is given by:    \$$T_n=a+(n-1)d\$$

\n

In this example \$$a=\\var{a}\$$,  \$$d = \\var{d}\$$  and  \$$n = \\var{n}\$$

\n

\$$T_\\var{n}=\\var{a}+\\simplify{{n}-1}*\\var{d}\$$

\n

\$$T_\\var{n}=\\var{a}+\\simplify{({n}-1)*{d}}\$$

\n

\$$T_\\var{n}=\\simplify{{a}+({n}-1)*{d}}\$$

", "variable_groups": [], "parts": [{"scripts": {}, "showFeedbackIcon": true, "prompt": "

Calculate the \$$\\var{n}th\$$ term of the series.

\n

\$$T_\\var{n}=\$$ [[0]]

", "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"minValue": "{a}+({n}-1)*{d}", "precision": "1", "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{a}+({n}-1)*{d}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true}], "statement": "

The first three terms of a series are given by:

\n

\$$\\var{a} + \\simplify{{a}+{d}} + \\simplify{{a}+2*{d}}\\,+ \\, ...........\$$

", "functions": {}, "preamble": {"js": "", "css": ""}, "rulesets": {}, "variables": {"n": {"definition": "random(4..19#1)", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(1..12#1)", "templateType": "randrange", "name": "a", "group": "Ungrouped variables", "description": ""}, "d": {"definition": "random(2..11#1)", "templateType": "randrange", "name": "d", "group": "Ungrouped variables", "description": ""}}, "ungrouped_variables": ["a", "d", "n"], "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "type": "question"}, {"name": "Arithmetic progression: The sum of the first n terms of a series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Find the sum of the first n terms of an arithmetic progression

"}, "advice": "

If the difference between successive pairs of terms is a constant then the series under examination is an arithmetic progression.

\n

Ths first term is \$$a\$$ and the common difference is \$$d\$$.

\n

The formula for the nth term of the series is given by:    \$$S_n=\\frac{n}{2}\\left(2a+(n-1)d\\right)\$$

\n

In this example \$$a=\\var{a}\$$,  \$$d = \\var{d}\$$  and  \$$n = \\var{n}\$$

\n

\$$S_\\var{n}=\\frac{\\var{n}}{2}\\left(2*\\var{a}+(\\var{n}-1)\\var{d}\\right)\$$

\n

\$$S_\\var{n}=\\simplify{{n}/{2}}\\left(\\simplify{2{a}}+\\simplify{({n}-1)*{d}}\\right)\$$

\n

\$$S_\\var{n}=\\simplify{{n}/{2}}\\left(\\simplify{2{a}+({n}-1)*{d}}\\right)\$$

\n

\$$S_\\var{n}=\\simplify{{n}*{a}+{n}*({n}-1)*{d}/2}\$$

\n

", "tags": [], "functions": {}, "statement": "

The first three terms of a series are given by:

\n

\$$\\var{a} + \\simplify{{a}+{d}} + \\simplify{{a}+2*{d}}\\,+ \\, ...........\$$

", "preamble": {"js": "", "css": ""}, "rulesets": {}, "variables": {"n": {"definition": "random(4..19#1)", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(1..12#1)", "templateType": "randrange", "name": "a", "group": "Ungrouped variables", "description": ""}, "d": {"definition": "random(2..11#1)", "templateType": "randrange", "name": "d", "group": "Ungrouped variables", "description": ""}}, "ungrouped_variables": ["a", "d", "n"], "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "prompt": "

Calculate the sum of the first \$$\\var{n}\$$ terms of this series.

\n

\$$S_\\var{n}=\$$ [[0]]

", "type": "gapfill", "marks": 0, "gaps": [{"minValue": "{n}*{a}+({n}-1)*{n}*{d}/2", "precision": "1", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{n}*{a}+({n}-1)*{n}*{d}/2", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "mustBeReduced": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0}], "variableReplacementStrategy": "originalfirst"}], "variable_groups": [], "type": "question"}, {"name": "Arithmetic progressions: simultaneous equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "statement": "

The sum of the first \$$\\var{n1}\$$ terms of an arithmetic progression is \$$\\var{s1}\$$ and the \$$\\var{n2}th\$$ term of the same series is \$$\\var{T}\$$.

", "tags": [], "preamble": {"js": "", "css": ""}, "variable_groups": [], "variables": {"T": {"definition": "random(8..16#1)", "description": "", "name": "T", "templateType": "randrange", "group": "Ungrouped variables"}, "d": {"definition": "({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))", "description": "", "name": "d", "templateType": "anything", "group": "Ungrouped variables"}, "n2": {"definition": "random(5..14#1)", "description": "", "name": "n2", "templateType": "randrange", "group": "Ungrouped variables"}, "n1": {"definition": "random(3..12#1)", "description": "", "name": "n1", "templateType": "randrange", "group": "Ungrouped variables"}, "s1": {"definition": "random(12..60#1)", "description": "", "name": "s1", "templateType": "randrange", "group": "Ungrouped variables"}, "a": {"definition": "{T}-({n2}-1)*{d}", "description": "", "name": "a", "templateType": "anything", "group": "Ungrouped variables"}}, "rulesets": {}, "advice": "

Recall the formula for the sum of the first n terms of an arithmetic progression is \$$S_n=\\frac{n}{2}(2a+(n-1)d)\$$.

\n

The sum of the first \$$\\var{n1}\$$ terms of an arithmetic progression is \$$\\var{s1}\$$

\n

\$$\\frac{\\var{n1}}{2}(2a+\\simplify{{n1}-1}d)=\\var{s1}\$$                               equation (i)

\n

The formula for nth term of an arithmetic progression is \$$T_n=a+(n-1)d\$$.

\n

The \$$\\var{n2}th\$$ term of the same series is \$$\\var{T}\$$

\n

\$$a+\\simplify{{n2}-1}d=\\var{T}\$$                                                   equation (ii)

\n

Here we have two simultaneous equations. We can eliminate the \$$a\$$ term.

\n

\$$\\var{n1}a+\\simplify{({n1}-1)*{n1}/2}d=\\var{s1}\$$                    equation (i)

\n

\$$\\var{n1}a+\\simplify{{n1}*({n2}-1)}d=\\simplify{{n1}*{T}}\$$                  equation (ii)*\$$\\var{n1}\$$

\n

\$$\\simplify{({n1}-1)*{n1}/2-{n1}*({n2}-1)}d=\\simplify{{s1}-{n1}*{T}}\$$

\n

\$$d=\\frac{\\simplify{{s1}-{n1}*{T}}}{\\simplify{({n1}-1)*{n1}/2-{n1}*({n2}-1)}}\$$

\n

\$$d=\\simplify{({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\$$

\n

Using this result and equation (ii) we can find the value for \$$a\$$

\n

\$$a+\\simplify{({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}=\\var{T}\$$

\n

\$$a=\\var{T}-\\simplify{({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\$$

\n

\$$a=\\simplify{{T}-({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\$$

\n

", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Solving arithmetic progressions using simultaneous equations

"}, "ungrouped_variables": ["n1", "s1", "n2", "T", "d", "a"], "parts": [{"prompt": "

Calculate the value of the common difference.   \$$d\$$ = [[0]]

\n

Calculate the value of the first term of the series.  \$$a\$$ = [[1]]

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "{d}", "correctAnswerFraction": false, "precisionPartialCredit": 0, "precisionType": "dp", "showFeedbackIcon": true, "type": "numberentry", "marks": 1, "variableReplacements": [], "mustBeReduced": false, "precision": "1", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "maxValue": "{d}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "allowFractions": false, "showPrecisionHint": true, "strictPrecision": false}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "{a}", "correctAnswerFraction": false, "precisionPartialCredit": 0, "precisionType": "dp", "showFeedbackIcon": true, "type": "numberentry", "marks": 1, "variableReplacements": [], "mustBeReduced": false, "precision": "1", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "maxValue": "{a}", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "allowFractions": false, "showPrecisionHint": true, "strictPrecision": false}], "variableReplacements": [], "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "scripts": {}}], "type": "question"}, {"name": "Arithmetic progressions: simultaneous equations #2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "statement": "

The sum of the first \$$\\var{n1}\$$ terms of an arithmetic progression is \$$\\var{s1}\$$ and the sum of the first \$$\\var{n2}\$$ terms of an arithmetic progression is \$$\\var{s2}\$$

", "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "tags": [], "variables": {"s1": {"description": "", "name": "s1", "definition": "random(12..60#1)", "templateType": "randrange", "group": "Ungrouped variables"}, "s2": {"description": "", "name": "s2", "definition": "random(48..90#1)", "templateType": "randrange", "group": "Ungrouped variables"}, "a": {"description": "", "name": "a", "definition": "({s1}-{n1}*({n1}-1)*d/2)/{n1}", "templateType": "anything", "group": "Ungrouped variables"}, "d": {"description": "", "name": "d", "definition": "2*({n2}*{s1}-{n1}*{s2})/({n1}*{n2}*({n1}-{n2}))", "templateType": "anything", "group": "Ungrouped variables"}, "n1": {"description": "", "name": "n1", "definition": "random(3..8#1)", "templateType": "randrange", "group": "Ungrouped variables"}, "n2": {"description": "", "name": "n2", "definition": "random(9..15#1)", "templateType": "randrange", "group": "Ungrouped variables"}}, "variable_groups": [], "metadata": {"description": "

Solving arithmetic progressions using simultaneous equations

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "parts": [{"marks": 0, "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "

Calculate the value of the common difference.   \$$d\$$ = [[0]]

\n

Calculate the value of the first term of the series.  \$$a\$$ = [[1]]

", "gaps": [{"mustBeReducedPC": 0, "marks": 1, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "{d}", "minValue": "{d}", "allowFractions": false, "precisionPartialCredit": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precisionType": "dp", "showPrecisionHint": true, "precision": "1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}}, {"mustBeReducedPC": 0, "marks": 1, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "{a}", "minValue": "{a}", "allowFractions": false, "precisionPartialCredit": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precisionType": "dp", "showPrecisionHint": true, "precision": "1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}}]}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["n1", "s1", "n2", "s2", "d", "a"], "advice": "

Recall the formula for the sum of the first n terms of an arithmetic progression is \$$S_n=\\frac{n}{2}(2a+(n-1)d)\$$.

\n

The sum of the first \$$\\var{n1}\$$ terms of an arithmetic progression is \$$\\var{s1}\$$

\n

\$$\\frac{\\var{n1}}{2}(2a+\\simplify{{n1}-1}d)=\\var{s1}\$$

\n

\$$\\var{n1}a+\\simplify{{n1}*({n1}-1)/2}d=\\var{s1}\$$                                       equation (i)

\n

The sum of the first \$$\\var{n2}\$$ terms of an arithmetic progression is \$$\\var{s2}\$$

\n

\$$\\frac{\\var{n2}}{2}(2a+\\simplify{{n2}-1}d)=\\var{s2}\$$

\n

\$$\\var{n2}a+\\simplify{{n2}*({n2}-1)/2}d=\\var{s2}\$$                                      equation (ii)

\n

We can eliminate the \$$a\$$ term.

\n

\$$\\simplify{{n2}*{n1}}a+\\simplify{{n2}*{n1}*({n1}-1)/2}d=\\simplify{{n2}*{s1}}\$$                               equation (i) * \$$\\var{n2}\$$

\n

\$$\\simplify{{n2}*{n1}}a+\\simplify{{n1}*{n2}*({n2}-1)/2}d=\\simplify{{n1}*{s2}}\$$                               equation (ii) * \$$\\var{n1}\$$

\n

Subtracting gives

\n

\$$\\simplify{{n2}*{n1}*({n1}-{n2})/2}d=\\simplify{{n2}*{s1}-{n1}*{s2}}\$$

\n

\$$d=\\frac{\\simplify{{n2}*{s1}-{n1}*{s2}}}{\\simplify{{n2}*{n1}*({n1}-{n2})/2}}\$$

\n

\$$d=\\simplify{2*({n2}*{s1}-{n1}*{s2})/({n2}*{n1}*({n1}-{n2}))}\$$

\n

Inserting this value in for \$$d\$$ in equation (i) gives

\n

\$$\\var{n1}a+\\simplify{{n1}*({n1}-1)/2}(\\simplify{2*({n2}*{s1}-{n1}*{s2})/({n2}*{n1}*({n1}-{n2}))})=\\var{s1}\$$

\n

\$$\\var{n1}a+(\\simplify{({n1}-1)*({n2}*{s1}-{n1}*{s2})/({n2}*({n1}-{n2}))})=\\var{s1}\$$

\n

\$$\\var{n1}a=\\var{s1}-(\\simplify{({n1}-1)*({n2}*{s1}-{n1}*{s2})/({n2}*({n1}-{n2}))})\$$

\n

\$$\\var{n1}a=\\simplify{{s1}-({n1}-1)*({n2}*{s1}-{n1}*{s2})/({n2}*({n1}-{n2}))}\$$

\n

\$$a=\\simplify{({s1}-({n1}-1)*({n2}*{s1}-{n1}*{s2})/({n2}*({n1}-{n2})))/{n1}}\$$

\n

\$$a=\\var{a}\$$

\n

", "type": "question"}, {"name": "Geometric progression: The nth term of a series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "variables": {"a": {"name": "a", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(1..12#1)"}, "n": {"name": "n", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(4..19#1)"}, "r": {"name": "r", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(0.2..3#0.2)"}}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "statement": "

The first three terms of a series are given by:

\n

\$$\\var{a} + \\simplify{{a}*{r}} + \\simplify{{a}*{r}^2}\\,+ \\, ...........\$$

", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Find the nth term of a Geometric progression

"}, "ungrouped_variables": ["a", "r", "n"], "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "type": "gapfill", "gaps": [{"showFeedbackIcon": true, "minValue": "{a}*r^({n}-1)", "variableReplacements": [], "correctAnswerStyle": "plain", "precisionType": "dp", "allowFractions": false, "precision": "1", "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "mustBeReduced": false, "mustBeReducedPC": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "maxValue": "{a}*r^({n}-1)", "marks": 1, "precisionPartialCredit": 0, "type": "numberentry", "showPrecisionHint": true, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"]}], "showCorrectAnswer": true, "prompt": "

Calculate the \$$\\var{n}th\$$ term of the series.

\n

\$$T_\\var{n}=\$$ [[0]]

", "scripts": {}}], "advice": "

If the ratio between successive pairs of terms is a constant then the series under examination is a geometric progression.

\n

Ths first term is \$$a\$$ and the common ratio is \$$r\$$.

\n

The formula for the nth term of the series is given by:    \$$T_n=ar^{n-1}\$$

\n

In this example \$$a=\\var{a}\$$,   \$$r = \\frac{\\simplify{{a}*{r}}}{\\var{a}}=\\var{r}\$$  and  \$$n = \\var{n}\$$

\n

\$$T_\\var{n}=\\var{a}*\\var{r}^{\\simplify{{n}-1}}\$$

\n

\$$T_\\var{n}=\\var{a}*\\simplify{{r}^{{n}-1}}\$$

\n

\$$T_\\var{n}=\\simplify{{a}*{r}^{{n}-1}}\$$

", "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "rulesets": {}, "tags": [], "type": "question"}, {"name": "Geometric progression: The sum of the first n terms of a geometric progression", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Find the sum of the first n terms of a Geometric progression

"}, "advice": "

If the ratio between successive pairs of terms is a constant then the series under examination is a geometric progression.

\n

Ths first term is \$$a\$$ and the common ratio is \$$r\$$.

\n

The formula for the sum of the first \$$n\$$ terms of the series is given by:    \$$S_n=\\frac{a(1-r^{n})}{1-r}\$$

\n

In this example \$$a=\\var{a}\$$,   \$$r = \\frac{\\simplify{{a}*{r}}}{\\var{a}}=\\var{r}\$$  and  \$$n = \\var{n}\$$

\n

\$$S_\\var{n}=\\frac{\\var{a}(1-(\\var{r})^{\\var{n}})}{1-\\var{r}}\$$

\n

\$$S_\\var{n}=\\frac{\\var{a}*(\\simplify{1-{r}^{n}})}{\\simplify{1-{r}}}\$$

\n

\$$S_\\var{n}=\\frac{\\simplify{{a}*(1-{r}^{n})}}{\\simplify{1-{r}}}\$$

\n

\$$S_\\var{n}=\\var{s}\$$

", "variable_groups": [], "parts": [{"showFeedbackIcon": true, "marks": 0, "scripts": {}, "variableReplacements": [], "prompt": "

Calculate the sum of the first \$$\\var{n}\$$ terms of the series.

\n

\$$S_\\var{n}=\$$ [[0]]

", "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"precision": "1", "minValue": "{s}", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{s}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "mustBeReduced": false, "type": "numberentry", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst"}], "statement": "

The first three terms of a series are given by:

\n

\$$\\var{a} + \\simplify{{a}*{r}} + \\simplify{{a}*{r}^2}\\,+ \\, ...........\$$

", "preamble": {"js": "", "css": ""}, "rulesets": {}, "variables": {"n": {"definition": "random(4..19#1)", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(1..12#1)", "templateType": "randrange", "name": "a", "group": "Ungrouped variables", "description": ""}, "s": {"definition": "({a}*(1-{r}^{n}))/(1-{r})", "templateType": "anything", "name": "s", "group": "Ungrouped variables", "description": ""}, "r": {"definition": "random(0.2..3#0.2)", "templateType": "randrange", "name": "r", "group": "Ungrouped variables", "description": ""}}, "ungrouped_variables": ["a", "r", "n", "s"], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "type": "question"}, {"name": "Solving for a geometric series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Solving for a geometric series

"}, "advice": "

\$$S_{\\var{n}}=\\frac{a(1-r^{\\var{n}})}{1-r}=\\var{s_1}\$$

\n

\$$S_{\\simplify{2*{n}}}=\\frac{a(1-r^{\\simplify{2*{n}}})}{1-r}=\\var{s_2}\$$

\n

If we divide one by the other we get:

\n

\$$\\frac{S_{\\simplify{2*{n}}}}{S_{\\var{n}}}=\\frac{\\frac{a(1-r^{\\simplify{2*{n}}})}{1-r}}{\\frac{a(1-r^{\\var{n}})}{1-r}}=\\frac{\\var{s_2}}{\\var{s_1}}\$$

\n

\$$\\frac{S_{\\simplify{2*{n}}}}{S_{\\var{n}}}=\\frac{a(1-r^{\\simplify{2*{n}}})}{1-r}*\\frac{1-r}{a(1-r^{\\var{n}})}=\\frac{\\var{s_2}}{\\var{s_1}}\$$

\n

\$$\\frac{1-r^{\\simplify{2*{n}}}}{1-r^{\\var{n}}}=\\frac{\\var{s_2}}{\\var{s_1}}\$$

\n

\$$\\frac{(1-r^\\var{n})(1+r^{\\var{n}})}{1-r^{\\var{n}}}=\\frac{\\var{s_2}}{\\var{s_1}}\$$

\n

\$$1+r^{\\var{n}}=\\frac{\\var{s_2}}{\\var{s_1}}\$$

\n

\$$r^{\\var{n}}=\\frac{\\var{s_2}}{\\var{s_1}}-1\$$

\n

\$$r^{\\var{n}}=\\simplify{{s_2}/{s_1}-1}\$$

\n

\$$r=\\simplify{(({s_2})/{s_1}-1)^{1/{n}}}\$$

\n

\$$r=\\simplify{(({s_2}-{s_1})/{s_1})^{1/{n}}}=\\var{r}\$$

\n

Recall \$$S_{\\var{n}}=\\frac{a(1-r^{\\var{n}})}{1-r}=\\var{s_1}\$$

\n

\$$a=\\frac{\\var{s_1}*(1-{r})}{1-r^{\\var{n}}}\$$

\n

Inserting the value for \$$r\$$ in this equation gives

\n

\$$a=\\frac{\\var{s_1}*(\\simplify{(1-{r})})}{\\simplify{{1-r^{{n}}}}}\$$

\n

\$$a=\\var{a}\$$

\n

", "tags": [], "functions": {}, "statement": "

The sum of the first \$$\\var{n}\$$ terms of a geometric series is \$$\\var{s_1}\$$ and the sum of the first \$$\\simplify{2*{n}}\$$ terms is \$$\\var{s_2}\$$.

", "preamble": {"js": "", "css": ""}, "rulesets": {}, "variables": {"n": {"definition": "random(2..6#1)", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "{s_1}*(1-{r})/(1-{r}^{n})", "templateType": "anything", "name": "a", "group": "Ungrouped variables", "description": ""}, "s_2": {"definition": "random(36..50#1)", "templateType": "randrange", "name": "s_2", "group": "Ungrouped variables", "description": ""}, "s_1": {"definition": "random(12..36#1)", "templateType": "randrange", "name": "s_1", "group": "Ungrouped variables", "description": ""}, "r": {"definition": "({s_2}/{s_1}-1)^(1/{n})", "templateType": "anything", "name": "r", "group": "Ungrouped variables", "description": ""}}, "ungrouped_variables": ["n", "s_1", "s_2", "r", "a"], "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "prompt": "

Determine the value of the common ratio.    \$$r\$$ = [[0]]

\n

Calculate the value of the first term.    \$$a\$$ = [[1]]

", "type": "gapfill", "marks": 0, "gaps": [{"minValue": "{r}", "precision": "2", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{r}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "mustBeReduced": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"minValue": "{a}", "precision": "2", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{a}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "mustBeReduced": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0}], "variableReplacementStrategy": "originalfirst"}], "variable_groups": [], "type": "question"}, {"name": "Solving for a geometric series #2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "rulesets": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "tags": [], "metadata": {"description": "

Solving for a geometric series

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "variables": {"r": {"description": "", "name": "r", "definition": "(1-s1/s2)^(1/{n})", "templateType": "anything", "group": "Ungrouped variables"}, "s1": {"description": "", "name": "s1", "definition": "random(12..50#1)", "templateType": "randrange", "group": "Ungrouped variables"}, "s2": {"description": "", "name": "s2", "definition": "2*{s1}-{n}", "templateType": "anything", "group": "Ungrouped variables"}, "a": {"description": "", "name": "a", "definition": "{s2}*(1-{r})", "templateType": "anything", "group": "Ungrouped variables"}, "n": {"description": "", "name": "n", "definition": "random(5..12#1)", "templateType": "randrange", "group": "Ungrouped variables"}}, "parts": [{"marks": 0, "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "

Determine the value of the common ratio.    \$$r\$$ = [[0]]

\n

Calculate the value of the first term.    \$$a\$$ = [[1]]

", "gaps": [{"marks": 1, "showPrecisionHint": true, "mustBeReduced": false, "maxValue": "{r}", "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "type": "numberentry", "scripts": {}, "allowFractions": false, "precisionPartialCredit": 0, "minValue": "{r}", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precisionType": "dp", "mustBeReducedPC": 0, "precision": "2", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision."}, {"marks": 1, "showPrecisionHint": true, "mustBeReduced": false, "maxValue": "{a}", "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "type": "numberentry", "scripts": {}, "allowFractions": false, "precisionPartialCredit": 0, "minValue": "{a}", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precisionType": "dp", "mustBeReducedPC": 0, "precision": "2", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision."}]}], "statement": "

The sum of the first \$$\\var{n}\$$ terms of a geometric series is \$$\\var{s1}\$$ and the sum to infinity of the series is \$$\\var{s2}\$$.

", "variable_groups": [], "functions": {}, "advice": "

\$$S_{\\var{n}}=\\frac{a(1-r^{\\var{n}})}{1-r}=\\var{s1}\$$

\n

\$$S_{\\infty}=\\frac{a}{1-r}=\\var{s2}\$$

\n

If we divide one by the other we get:

\n

\$$\\frac{S_{\\simplify{n}}}{S_{\\infty}}=\\frac{\\frac{a(1-r^{\\simplify{n}})}{1-r}}{\\frac{a}{1-r}}=\\frac{\\var{s1}}{\\var{s2}}\$$

\n

\$$\\frac{S_{\\simplify{n}}}{S_{\\infty}}=\\frac{a(1-r^{\\simplify{n}})}{1-r}*\\frac{1-r}{a}=\\frac{\\var{s1}}{\\var{s2}}\$$

\n

\$$1-r^{{n}}=\\frac{\\var{s1}}{\\var{s2}}\$$

\n

\$$1-\\frac{\\var{s1}}{\\var{s2}}=r^{{n}}\$$

\n

In this example \$$n=\\var{n}\$$

\n

\$$r^{\\var{n}}=\\simplify{({s2}-{s1})/{s2}}\$$

\n

\$$r=(\\simplify{({s2}-{s1})/{s2}})^{1/\\var{n}}\$$

\n

\$$r=\\simplify{(({s2}-{s1})/{s1})^{1/{n}}}=\\var{r}\$$

\n

Recall \$$S_{\\infty}=\\frac{a}{1-r}=\\var{s2}\$$

\n

\$$a=\\var{s2}*(1-{r})\$$

\n

Inserting the value for \$$r\$$ in this equation gives

\n

\$$a=\\var{s2}*\\simplify{(1-{r})}\$$

\n

\$$a=\\var{a}\$$

\n

", "ungrouped_variables": ["n", "s1", "s2", "r", "a"], "type": "question"}, {"name": "Solving for a geometric series #3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Solving for a geometric series

"}, "advice": "

The second term of a geometric series is given by the formula \$$T_2=ar\$$ and the sum to infinity of a geometric series is \$$S_\\infty=\\frac{a}{1-r}\$$

\n

\$$T_2=ar=\\var{t2}\$$

\n

\$$a=\\frac{\\var{t2}}{r}\$$

\n

We can substitute this in for \$$a\$$ in the second equation

\n

\$$S_\\infty=\\frac{a}{1-r}=\\var{s}\$$

\n

\$$\\frac{\\frac{\\var{t2}}{r}}{1-r}=\\var{s}\$$

\n

\$$\\frac{\\var{t2}}{r}=\\var{s}(1-{r})\$$

\n

\$$\\frac{\\var{t2}}{r}=\\var{s}-\\var{s}{r}\$$

\n

\$$\\var{t2}=\\var{s}r-\\var{s}r^2\$$

\n

\$$\\var{s}r^2-\\var{s}r+\\var{t2}=0\$$

\n

This is a quadratic equation which we can solve by formula.

\n

\$$r=\\frac{\\var{s}\\pm \\sqrt{(-\\var{s})^2-4*(\\var{s})*(\\var{t2})}}{2*(\\var{s})}\$$

\n

\$$r=\\frac{\\var{s}+\\sqrt{\\simplify{{s}^2-4*{s}*{t2}}}}{\\simplify{2*{s}}}\$$   or    \$$r=\\frac{\\var{s}-\\sqrt{\\simplify{{s}^2-4*{s}*{t2}}}}{\\simplify{2*{s}}}\$$

\n

\$$r=\\frac{\\var{s}+\\simplify{({s}^2-4*{s}*{t2})^0.5}}{\\simplify{2*{s}}}\$$    or    \$$r=\\frac{\\var{s}-\\simplify{({s}^2-4*{s}*{t2})^0.5}}{\\simplify{2*{s}}}\$$

\n

\$$r=\$$ {({s}+({s}^2-4*{s}*{t2})^0.5)/(2*{s})}   or    \$$r=\$$ {({s}-({s}^2-4*{s}*{t2})^0.5)/(2*{s})}

\n

\$$a=\\frac{\\var{t2}}{r}\$$

\n

\$$a=\$$ {(2*{s}*{t2})/({s}+({s}^2-4*{s}*{t2})^0.5)}    or    \$$a=\$$ {(2*{s}*{t2})/({s}-({s}^2-4*{s}*{t2})^0.5)}

\n

\n

\n

\n

\n

", "variable_groups": [], "parts": [{"scripts": {}, "showFeedbackIcon": true, "prompt": "

Calculate the value of the larger common ratio.   \$$r\$$ = [[0]]

\n

Determine the value of the first term of the series corresponding to this common ratio.  \$$a\$$ = [[1]]

\n

Calculate the value of the smaller common ratio.   \$$r\$$ = [[2]]

\n

Determine the value of the first term of the series corresponding to this common ratio.  \$$a\$$ = [[3]]

", "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"minValue": "{r_1}", "precision": "2", "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{r_1}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}, {"minValue": "{a_1}", "precision": "2", "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{a_1}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}, {"minValue": "{r_2}", "precision": "2", "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{r_2}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}, {"minValue": "{a_2}", "precision": "2", "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "showPrecisionHint": true, "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "maxValue": "{a_2}", "correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": false, "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true}], "statement": "

The second term in a geometric series is \$$\\var{t2}\$$ and the sum to infinity of the series is \$$\\var{s}\$$.

\n

There are two possible series that possess these attributes.

", "functions": {}, "preamble": {"js": "", "css": ""}, "rulesets": {}, "variables": {"a_2": {"definition": "{t2}/{r_2}", "templateType": "anything", "name": "a_2", "group": "Ungrouped variables", "description": ""}, "r_2": {"definition": "({s}-sqrt({s}^2-4*{s}*{t2}))/(2*{s})", "templateType": "anything", "name": "r_2", "group": "Ungrouped variables", "description": ""}, "t2": {"definition": "random(1..9#1)", "templateType": "randrange", "name": "t2", "group": "Ungrouped variables", "description": ""}, "a_1": {"definition": "{t2}/{r_1}", "templateType": "anything", "name": "a_1", "group": "Ungrouped variables", "description": ""}, "s": {"definition": "random(36..50#1)", "templateType": "randrange", "name": "s", "group": "Ungrouped variables", "description": ""}, "r_1": {"definition": "({s}+sqrt({s}^2-4*{s}*{t2}))/(2*{s})", "templateType": "anything", "name": "r_1", "group": "Ungrouped variables", "description": ""}}, "ungrouped_variables": ["t2", "s", "r_1", "r_2", "a_1", "a_2"], "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "type": "question"}, {"name": "First 3 terms of Binomial series for Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "variables": {"k": {"description": "", "templateType": "anything", "name": "k", "group": "Ungrouped variables", "definition": "random(2..{n}-1)"}, "a": {"description": "", "templateType": "randrange", "name": "a", "group": "Ungrouped variables", "definition": "random(2..10#1)"}, "n": {"description": "", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "definition": "random(4..9#1)"}, "c": {"description": "", "templateType": "anything", "name": "c", "group": "Ungrouped variables", "definition": "comb({n},{k})*{a}^({n}-{k})*{b}^{k}"}, "b": {"description": "", "templateType": "randrange", "name": "b", "group": "Ungrouped variables", "definition": "random(2..12#1)"}}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "statement": "

Given the expression \$$(\\var{a}+\\var{b}x)^{\\var{n}}\$$

", "metadata": {"description": "

Find the first 3 terms of Binomial series having a Natural exponent

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "advice": "

The binomial series expansion for an expression of the form \$$(a+bx)^n\$$ where \$$n\$$ is a Natural number is given by:

\n

\$$(a+bx)^n=\\tbinom{n}{0}(a)^n(bx)^{0}+\\tbinom{n}{1}(a)^{n-1}(bx)^{1}+\\tbinom{n}{2}(a)^{n-2}(bx)^{2}+...\\tbinom{n}{k}(a)^{n-k}(bx)^{k}+...\\tbinom{n}{n}(a)^{0}(bx)^{n}\$$

\n

In this example  \$$n=\\var{n}\$$,  \$$k=\\var{k}\$$,  \$$a=\\var{a}\$$  and  \$$b=\\var{b}\$$.

\n

So the first three terms of the binomial series expansion are:

\n

\$$\\var{a}^{\\var{n}}+\\tbinom{\\var{n}}{\\var{1}}*\\var{a}^{\\var{n}-1}*\\var{b}^{1}+\\tbinom{\\var{n}}{2}*\\var{a}^{\\var{n}-2}*\\var{b}^{2}\$$

\n

\$$=\\simplify{{a}^{n}}+\\simplify{{n}*{a}^({n}-1)*{b}}x+\\simplify{{n}*{n-1}*{a}^{{n}-2}*{b}^2/2x^2}\$$

\n

", "parts": [{"showFeedbackIcon": true, "marks": "3", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "prompt": "

Write out the first three terms of the binomial expansion.

", "scripts": {}, "answer": "{a}^{n}+{n}*{a}^({n}-1)*{b}x+{n}*{n-1}*{a}^({n}-2)*{b}^2/2x^2", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff", "expectedvariablenames": [], "type": "jme", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showpreview": true}], "ungrouped_variables": ["a", "b", "n", "c", "k"], "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "rulesets": {}, "functions": {}, "type": "question"}, {"name": "Binomial series for Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "tags": [], "statement": "

Given the expression \$$(\\var{a}+\\var{b}x)^{\\var{n}}\$$

", "parts": [{"prompt": "

By using the binomial series expansion, calculate the coefficient of \$$x^{\\var{k}}\$$  [[0]]

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "allowFractions": true, "showFeedbackIcon": true, "type": "numberentry", "marks": 1, "variableReplacements": [], "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "maxValue": "{c}", "mustBeReducedPC": 0, "minValue": "{c}", "scripts": {}}], "variableReplacements": [], "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "scripts": {}}], "variable_groups": [], "variables": {"k": {"definition": "random(2..{n}-1)", "description": "", "name": "k", "templateType": "anything", "group": "Ungrouped variables"}, "b": {"definition": "random(2..12#1)", "description": "", "name": "b", "templateType": "randrange", "group": "Ungrouped variables"}, "a": {"definition": "random(2..10#1)", "description": "", "name": "a", "templateType": "randrange", "group": "Ungrouped variables"}, "c": {"definition": "comb({n},{k})*{a}^({n}-{k})*{b}^{k}", "description": "", "name": "c", "templateType": "anything", "group": "Ungrouped variables"}, "n": {"definition": "random(4..9#1)", "description": "", "name": "n", "templateType": "randrange", "group": "Ungrouped variables"}}, "rulesets": {}, "advice": "

The binomial series expansion for an expression of the form \$$(a+bx)^n\$$ where \$$n\$$ is a Natural number is given by:

\n

\$$(a+bx)^n=\\tbinom{n}{0}(a)^n(bx)^{0}+\\tbinom{n}{1}(a)^{n-1}(bx)^{1}+\\tbinom{n}{2}(a)^{n-2}(bx)^{2}+...\\tbinom{n}{k}(a)^{n-k}(bx)^{k}+...\\tbinom{n}{n}(a)^{0}(bx)^{n}\$$

\n

In this example  \$$n=\\var{n}\$$,  \$$k=\\var{k}\$$,  \$$a=\\var{a}\$$  and  \$$b=\\var{b}\$$.

\n

So the coefficient of \$$x^{\\var{k}}\$$ is given by \$$\\tbinom{\\var{n}}{\\var{k}}*\\var{a}^{\\var{n}-\\var{k}}*\\var{b}^{\\var{k}}=\\var{c}\$$.

\n

", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Binomial series for Natural exponent

"}, "ungrouped_variables": ["a", "b", "n", "c", "k"], "functions": {}, "type": "question"}, {"name": " Binomial series for non-Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "ungrouped_variables": ["b", "n", "T1", "T2", "T3"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Find the first three terms in the binomial series expansion having a non-Natural exponent

"}, "tags": [], "functions": {}, "parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "checkvariablenames": false, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "answer": "{T1}+{T2}x+{T3}x^2", "prompt": "

Input the first three tems in the binomial series expansion.

", "variableReplacements": [], "scripts": {}, "marks": "3", "expectedvariablenames": []}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The binomial series expansion for an expression of the form \$$(1+bx)^{n}\$$ where \$$n\$$ is a non-natural number, is given by:

\n

\$$(1+bx)^{n}=1+\\tbinom{n}{1}(bx)^{1}+\\tbinom{{n}}{2}(bx)^{2}+\\tbinom{{n}}{3}(bx)^{3}.......\$$

\n

In this example  \$$power=\\frac{1}{\\var{n}}\$$,   and  \$$b=\\var{b}\$$.

\n

So the first three terms are

\n

\$$(1+\\var{b}x)^{\\frac{1}{\\var{n}}}=1+\\tbinom{\\frac{1}{\\var{n}}}{1}(\\var{b}x)^{1}+\\tbinom{\\frac{1}{\\var{n}}}{2}(\\var{b}x)^{2}\$$

\n

\$$(1+\\var{b}x)^{\\frac{1}{\\var{n}}}=\\var{T1}+\\var{T2}x+\\var{T3}x^2\$$

", "variable_groups": [], "statement": "

Given the expression \$$(1+\\var{b}x)^{\\frac{1}{\\var{n}}}\$$

", "variables": {"n": {"name": "n", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(2..6#1)"}, "T2": {"name": "T2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "{b}/{n}"}, "b": {"name": "b", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(2..12#1)"}, "T1": {"name": "T1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "1"}, "T3": {"name": "T3", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "{b}^2*(1/{n}*(1/{n}-1))/2"}}, "preamble": {"css": "", "js": ""}, "type": "question"}]}], "feedback": {"showanswerstate": true, "advicethreshold": 0, "allowrevealanswer": true, "showactualmark": true, "feedbackmessages": [], "showtotalmark": true, "intro": ""}, "percentPass": 0, "navigation": {"preventleave": true, "onleave": {"message": "", "action": "none"}, "showresultspage": "oncompletion", "reverse": true, "allowregen": true, "showfrontpage": true, "browse": true}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

This quiz poses questions on arithmetic progressions, geometric progressions and binomial series expansions.

"}, "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "extensions": [], "custom_part_types": [], "resources": []}