// Numbas version: exam_results_page_options {"metadata": {"description": "

Questions on arithmetic, or linear, sequences.

", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "name": "Arithmetic sequences", "type": "exam", "navigation": {"browse": true, "allowregen": true, "preventleave": true, "showfrontpage": true, "onleave": {"action": "none", "message": ""}, "reverse": true, "showresultspage": "oncompletion"}, "showQuestionGroupNames": false, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": "Write down and apply the formula for an arithmetic sequence.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "type": "jme", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "answer": "{a1}+(n-1){d}", "showFeedbackIcon": true, "answersimplification": "basic", "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "vsetrangepoints": 5, "expectedvariablenames": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "prompt": "

Write out an expression for $a_n$, the $n^{\\text{th}}$ term of the sequence, in terms of $n$.

\n

$a_n =$ [[0]]

\n

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\n

Find the $\\var{small}^{\\text{th}}$ term

\n

$a_{\\var{small}} =$ [[0]]

\n

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Find the $\\var{large}^{\\text{th}}$ term

\n

$a_{\\var{large}} =$[[0]]

Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.

"}, "tags": ["arithmetic sequences", "nth term", "sequences", "taxonomy"], "variables": {"large": {"templateType": "anything", "description": "

A large index to compute

", "definition": "random(10..50#5)*10", "name": "large", "group": "Ungrouped variables"}, "small": {"templateType": "anything", "description": "

A small index to compute

", "definition": "random(6..10)", "name": "small", "group": "Ungrouped variables"}, "a1": {"templateType": "anything", "description": "

The first term in the sequence

", "definition": "random(1..90)", "name": "a1", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "description": "", "definition": "random(3..13 except 10)", "name": "d", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["a1", "d", "small", "large"], "statement": "

In this question, consider the sequence

\n

\$a = \\var{a1}, \\; \\var{a1+d}, \\; \\var{a1+d*2}, \\; \\var{a1+d*3}, \\; \\ldots \$

\n

A helpful person has drawn out a table of the terms so far.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\boldsymbol{n}$ $\\boldsymbol{a_n}$ $1$ $2$ $3$ $4$ $\\ldots$ $\\var{a1}$ $\\var{a1+d}$ $\\var{a1+2d}$ $\\var{a1+3d}$ $\\ldots$

The formula for the $n^\\text{th}$ term, $a_n$, of an arithmetic sequence is

\n

\$a_n=a_1+(n-1)d \\text{.} \$

\n

$a_1$ is the first term, and $d$ is the common difference between adjacent terms.

\n

#### a)

\n

In the given sequence, the common difference is $\\var{a1+d} - \\var{a1} = \\var{d}$, and the first term is $\\var{a1}$.

\n

So, the formula for this sequence is

\n

\$a_n = \\var{a1} + (n-1) \\times \\var{d} \\text{.} \$

\n

#### b)

\n

\$a_\\var{small} = \\var{a1} + (\\var{small}-1) \\times \\var{d} = \\var{a1+(small-1)*d} \\text{.} \$

\n

#### c)

\n

\$a_\\var{large} = \\var{a1} + (\\var{large}-1) \\times \\var{d} = \\var{a1+(large-1)*d} \\text{.} \$

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$\\var{m[0]*(index[0])}, \\; \\var{m[0]*(index[0]+1)}, \\; \\var{m[0]*(index[0]+2)}, \\;$ [[0]] $, \\; \\var{m[0]*(index[0]+4)}, \\;$ [[1]]$, \\; \\var{m[0]*(index[0]+6)}$

\n

"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{m[1]*(index[1]-2)}", "maxValue": "{m[1]*(index[1]-2)}", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "{m[1]*(index[1]-5)}", "maxValue": "{m[1]*(index[1]-5)}", "marks": 1, "variableReplacements": []}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "

\n

$\\var{m[1]*(index[1])}, \\; \\var{m[1]*(index[1]-1)}, \\;$ [[0]]$, \\; \\var{m[1]*(index[1]-3)}, \\; \\var{m[1]*(index[1]-4)}, \\;$ [[1]] $, \\; \\var{m[1]*(index[1]-6)}$

In an arithmetic sequence, the difference between two adjecent terms is always the same.

\n

#### a)

\n

The difference between the first two terms in this sequence is $\\var{m[0]*(index[0]+1)} - \\var{m[0]*(index[0])} = \\var{m[0]}$.

\n

So to get from one term to the next, add $\\var{m[0]}$.

\n

The complete sequence is

\n

\$\\var{m[0]*(index[0])}, \\; \\var{m[0]*(index[0]+1)}, \\; \\var{m[0]*(index[0]+2)}, \\; \\underline{\\var{m[0]*(index[0]+3)}}, \\; \\var{m[0]*(index[0]+4)}, \\; \\underline{\\var{m[0]*(index[0]+5)}}, \\; \\var{m[0]*(index[0]+6)} \$

\n

#### b)

\n

The difference between the first two terms in this sequence is $\\var{m[1]*(index[1]-1)} - \\var{m[1]*(index[1])} = \\var{-m[1]}$.

\n

So to get from one term to the next, subtract $\\var{m[1]}$.

\n

The complete sequence is

\n

\$\\var{m[1]*(index[1])}, \\; \\var{m[1]*(index[1]-1)}, \\; \\underline{\\var{m[1]*(index[1]-2)}}, \\; \\var{m[1]*(index[1]-3)}, \\; \\var{m[1]*(index[1]-4)}, \\; \\underline{\\var{m[1]*(index[1]-5)}}, \\; \\var{m[1]*(index[1]-6)} \$

\n

\n

\n", "tags": ["Arithmetic sequences", "arithmetic sequences", "sequences", "taxonomy"], "variables": {"m": {"templateType": "anything", "description": "

Common differences of the sequences

", "definition": "shuffle(5..12 except 10)[0..2]", "name": "m", "group": "Ungrouped variables"}, "index": {"templateType": "anything", "description": "

Starting indices of the sequences.

", "definition": "shuffle(7..15 except 10)[0..2]", "name": "index", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["m", "index"], "statement": "

Fill in the gaps in the following arithmetic sequences.

Given arithmetic sequences with some terms missing, fill in the missing terms.

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$\\var{n[0]*index[0]}$, $\\var{n[0]*(index[0]+1)}$, $\\var{n[0]*(index[0]+2)}$, $\\var{n[0]*(index[0]+3)}$

\n

Common Difference = [[0]]

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$\\var{n[1]*(index[1])}$, $?$, $\\var{n[1]*(index[1]+2)}$, $?$, $\\var{n[1]*(index[1]+4)}$

\n

Common Difference = [[0]]

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$\\var{n[2]*(index[2])}$, $?$, $?$, $\\var{n[2]*(index[2]+3)}$, $?$, $?$, $\\var{n[2]*(index[2]+6)}$

\n

Common Difference = [[0]]

Given sequences with missing terms, find the common difference between terms.

"}, "tags": ["Arithmetic Sequences", "Arithmetic sequences", "arithmetic sequences", "common difference", "sequences", "taxonomy"], "variables": {"n": {"templateType": "anything", "description": "", "definition": "shuffle(3..9)[0..3]", "name": "n", "group": "Ungrouped variables"}, "index": {"templateType": "anything", "description": "", "definition": "shuffle(5..15 except 10)[0..3]", "name": "index", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["n", "index"], "statement": "

Find the common differences of the following arithmetic sequences. Some of the terms are missing.

In an arithmetic sequence, the difference between two adjecent terms is always the same.

\n

Call the common difference $d$, and the first term in the sequence $a_0$. Then the sequence goes as follows:

\n

\$a_0, \\; a_0+d, \\; a_0+2d, \\; a_0+3d, \\; \\ldots \$

\n

The difference between a term in the sequence and the term $n$ places along is $n \\times d$.

\n

#### a)

\n

The difference between the first two terms is $\\var{n[0]*(index[0]+1)} - \\var{n[0]*index[0]} = \\var{n[0]}$.

\n

So the common difference is $\\var{n[0]}$.

\n

#### b)

\n

We're not given any adjacent terms of this sequence, but we are given some terms two palces apart.

\n

\\begin{align}
2d &= \\var{n[1]*(index[1]+2)} - \\var{n[1]*index[1]} \\\\
&= \\var{2*n[1]} \\\\
d &= \\var{n[1]}
\\end{align}

\n

The common difference is $\\var{n[1]}$.

\n

#### c)

\n

Again we're not given any adjacent terms of this sequence, but we have two terms three places apart.

\n

\\begin{align}
3d &= \\var{n[2]*(index[2]+3)} - \\var{n[2]*index[2]} \\\\
&= \\var{3*n[2]} \\\\
d &= \\var{n[2]}
\\end{align}

\n

The common difference is $\\var{n[2]}$.

\n", "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Partial sum of an arithmetic sequence - birthday money", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "variable_groups": [{"variables": ["person", "pronouns"], "name": "A person"}], "functions": {}, "rulesets": {}, "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b", "first"], "metadata": {"description": "

The amount of money a person gets on their birthday follows an arithmetic sequence.

\n

Calculate the amount on a given birthday, then calculate the sum up to that point.

We are told that {person['name']}'s parents deposit a uniformly increasing amount of money into a savings account for {person['name']} every year on {person['name']}'s birthday.

\n

We are also given the amount of money that {person['pronouns']['their']} parents deposit into the account on {person['pronouns']['their']} first $3$ birthdays:

\n
\n
• On {person['pronouns']['their']} $1^{st}$ birthday, they deposited $£\\var{first}$ into the account.
• \n
• On {person['pronouns']['their']} $2^{nd}$ birthday, they deposited $£\\var{b[1]+first}$ into the account.
• \n
• On {person['pronouns']['their']} $3^{rd}$ birthday, they deposited $£\\var{b[1]*2+first}$ into the account.
• \n
\n

#### a)

\n

To calculate the amount of money {person['name']}'s parents would deposit into the savings account on {person['pronouns']['their']} 21st birthday, if {pronouns['their']} parents maintained this pattern, we use the equation

\n

\$a_n=a_1+(n-1)d\\text{,}\$

\n

where

\n
\n
• $a_1$ is the first term;
• \n
• $a_n$ is the $n^{\\text{th}}$ term;
• \n
• $d$ is the common difference between consecutive terms.
• \n
\n

To identify the first term and common difference of the sequence we can use a table like the one below.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$n$ $a_n$ First differences $1$ $2$ $3$ $\\mathbf{\\var{first}}$ $\\var{b[1]+first}$ $\\var{b[1]*2+first}$ $\\mathbf{\\var{b[1]}}$ $\\mathbf{\\var{b[1]}}$
\n

The first term and common difference have been highlighted in bold: $a_1 = \\var{first}$ and $d = \\var{b[1]}$.

\n

Now we can use these to calculate $a_{21}$, giving us

\n

\\begin{align}
a_{21}&=\\var{first}+\\var{b[1]} \\times (21-1) \\\\
&=\\var{first+b[1]*(20)}\\text{.} \\\\
\\end{align}

\n

So, assuming that {person['name']}'s parents do maintain this pattern, on {pronouns['their']} 21st birthday {pronouns['their']} parents will deposit $£\\var{first+b[1]*(20)}$ into the savings account.

\n

#### b)

\n

We are now asked to calculate the total amount of money that {person['name']}'s parents will have added to this savings account over 21 years, including the money that {pronouns['their']} parents will deposit into the account on {pronouns['their']} 21st birthday.

\n

This question involves calculating the sum using the equation

\n

\$\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\$

\n

We know from part a) that

\n

\\begin{align}
a_1&=\\var{first},\\\\
n&=21,\\\\
a_{21}&= \\var{first+b[1]*(20)}.
\\end{align}

\n

Using our formula for the sum,

\n

\\begin{align}
\\sum\\limits_{i=1}^n{a_i}&=\\frac{n}{2}(a_1+a_n)\\\\
&=\\frac{\\var{21}}{2}(\\var{first}+\\var{first+b[1]*(21-1)})\\\\
&=\\var{21*(first+first+b[1]*(20))/2}\\text{.}
\\end{align}

\n

Therefore, over 21 years {person['name']}'s parents will have added a total of $£\\var{21*(first+first+b[1]*(20))/2}$ to this savings account!

", "statement": "

{person['name']}'s parents deposit a uniformly increasing amount of money into a savings account for {pronouns['them']} every year on {pronouns['their']} birthday:

\n
\n
• on {pronouns['their']} first birthday {pronouns['their']} parents deposited $£\\var{first}$ into the account;
• \n
• on {pronouns['their']} second birthday {person['pronouns']['their']} parents deposited $£\\var{b[1]+first}$ into the account;
• \n
• on {pronouns['their']} third birthday {pronouns['their']} parents deposited $£\\var{b[1]*2+first}$ into the account.
• \n
\n

{person['name']} wants to know the total amount of money that will be in this savings account, excluding interest, after {pronouns['their']} 21st birthday, if {pronouns['their']} parents maintain this pattern.

", "preamble": {"js": "", "css": ""}, "variables": {"c": {"name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(3..13 except[10]),8)"}, "n": {"name": "n", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(3..9),7)"}, "person": {"name": "person", "description": "

A random person

", "templateType": "anything", "group": "A person", "definition": "random_person()"}, "m": {"name": "m", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(2..10),5)"}, "first": {"name": "first", "description": "

first term in the sequence

", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..15 #5)"}, "pronouns": {"name": "pronouns", "description": "", "templateType": "anything", "group": "A person", "definition": "person['pronouns']"}, "ni": {"name": "ni", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(19..40),10)"}, "b": {"name": "b", "description": "

a

", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(10..25 #5), 3)"}, "ci": {"name": "ci", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(6..20),10)"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "prompt": "

How much money will {person['name']}'s parents deposit into the savings account on {pronouns['their']} 21st birthday, if {pronouns['their']} parents maintain this pattern?

\n

£[[0]].

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Use the arithmetic formula,

\n

\$a_n = a_1 + (n-1)d, \$

\n

where

\n
\n
• $a_n$ is the $n^\\text{th}$ term;
• \n
• $a_1$ is the first term in the sequence;
• \n
• $d$ is the common difference between consecutive terms.
• \n
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What is the value of $a_1$?

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What is the value of $d$?

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Now use the formula to calculate $a_{21}$.

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "prompt": "

How much money will {person['name']}'s parents have added to this savings account over $21$ years in total, including the money that {person['pronouns']['their']} parents will deposit into the account on {person['pronouns']['their']} $21^{st}$ birthday?

\n

£[[0]].

\n

", "stepsPenalty": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{21*(first+first+b[1]*(20))/2}", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "{21*(first+first+b[1]*(20))/2}", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 1, "scripts": {}}], "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The sum of an arithmetic sequence $a_1, a_2, \\ldots, a_n$ is calculated by the following formula.

\n

\$\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\$

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}], "tags": ["Arithmetic sequences", "Arithmetic Sequences", "arithmetic sequences", "nth term", "partial sums", "random names", "sequences", "taxonomy"], "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Arithmetic sequences in an ice cream shop", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "metadata": {"description": "

Given the common difference and first term of an arithmetic sequence, work out the index of the nth term of the sequence.

\n

Framed as a word problem with ticket numbers in an ice cream shop.

", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": ["index", "d", "person"], "advice": "

We know that every $\\var{d}^{\\text{th}}$ ticket after the first receives strawberry ice cream. So, the sequence of ticket numbers which get strawberry ice cream starts like this:

\n

\$1, \\var{d+1}, \\var{2d+1}, \\var{3d+1}, \\ldots \$

\n

The numbers on the tickets for strawberry ice cream form an arithmetic sequence: the first term is $1$ and the common difference is $\\var{d}$.

\n

We can write down a formula for the $n^{\\text{th}}$ term in this sequence and rearrange it to find how many customers received strawberry ice cream before {person['name']}.

\n

The general formula for an arithmetic sequence is

\n

\$a_n = a_1 + (n-1)d \$

\n

where

\n
\n
• $a_n$ is the $n^{th}$ term;
• \n
• $a_1$ is the first term;
• \n
• $d$ is the common difference between consecutive terms.
• \n
\n

We know that $a_n = \\var{1+d*index}$, $a_1 = 1$, and $d = \\var{d}$, and we want to find $n$.

\n

Substituting these values from the formula gives

\n

\\begin{align}
a_n &= a_1 + (n-1)d \\\\
\\var{1+d*index} &= 1 + (n -1)\\var{d}\\,.
\\end{align}

\n

Now we rearrange this to find $n$:

\n

\\begin{align}
\\var{1+d*index-1} &= \\var{d}n - \\var{d} \\\\
\\var{d*index +d} &= \\var{d}n \\\\
n &= \\var{(d*index + d)/d}.
\\end{align}

\n

This is the number of people who received strawberry ice cream up to and including {person['name']}. Removing {person['name']} leaves $\\var{index}$ customers who received strawberry ice cream before {person['name']} did.

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

When customers enter an ice cream shop they receive a numbered ticket for a free sample.

\n

There are $\\var{d}$ flavours of ice cream that the shop alternates through sequentially.

\n

The first ticket was number $1$ and the person with this ticket received strawberry ice cream.

\n

{person['name']} was given ticket number $\\var{1+d*(index)}$ and also received strawberry ice cream.

", "parts": [{"correctAnswerFraction": false, "stepsPenalty": 0, "mustBeReducedPC": 0, "prompt": "

How many customers before {person['name']} have tried the strawberry ice-cream?

", "showFeedbackIcon": true, "allowFractions": false, "minValue": "index", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "index", "mustBeReduced": false, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "prompt": "

Use the arithmetic formula,

\n

\$a_n = a_1 + (n-1)d \$

\n

where
$a_n$ - The n$^{th}$ term in an arithmetic sequence
$a_1$ - The $1^{st}$ term in an arithmetic sequence
$n$ - Term number
$d$ - The common difference.

", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "prompt": "

For this arithmetic sequence, what is $a_1$?

", "showFeedbackIcon": true, "allowFractions": false, "minValue": "1", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "1", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain"}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "prompt": "

What is $d$?

", "showFeedbackIcon": true, "allowFractions": false, "minValue": "{d}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{d}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain"}], "marks": "3", "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "tags": ["arithmetic sequence", "calculate the term number", "common difference", "nth term", "sequences", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"person": {"description": "", "group": "Ungrouped variables", "definition": "random_person()", "name": "person", "templateType": "anything"}, "index": {"description": "", "group": "Ungrouped variables", "definition": "random(14..34 except [20, 21, 22, 30, 31, 32, 40])", "name": "index", "templateType": "anything"}, "d": {"description": "", "group": "Ungrouped variables", "definition": "random(6..12 except 10)", "name": "d", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Finding the formula for the $n^{\\text{th}}$ term of linear sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "type": "question", "statement": "

A linear sequence is a series of numbers that either increases or decreases by a constant amount at each step.

\n

Find formulas for the $n^{\\text{th}}$ term for each of the following linear sequences, where the values for $n=1\\text{,}2\\text{,}3$ are given:

", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"ni": {"group": "Ungrouped variables", "name": "ni", "description": "", "templateType": "anything", "definition": "repeat(random(19..40),10)"}, "c": {"group": "Ungrouped variables", "name": "c", "description": "", "templateType": "anything", "definition": "repeat(random(3..13 except[10]),8)"}, "n": {"group": "Ungrouped variables", "name": "n", "description": "", "templateType": "anything", "definition": "repeat(random(1..4),7)"}, "m": {"group": "Ungrouped variables", "name": "m", "description": "", "templateType": "anything", "definition": "repeat(random(2..10),5)"}, "b": {"group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything", "definition": "repeat(random(2..4), 5)"}, "ci": {"group": "Ungrouped variables", "name": "ci", "description": "", "templateType": "anything", "definition": "repeat(random(6..20),10)"}}, "functions": {}, "tags": ["arithmetic formula", "common difference", "first term", "formula for the nth term", "linear sequences", "nth term", "sequences", "taxonomy"], "variable_groups": [], "parts": [{"variableReplacements": [], "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answer": "{m[1]}*n+{m[1]}", "marks": "3", "variableReplacements": [], "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "

$\\var{m[1]*2}, \\var{m[1]*3}, \\var{m[1]*4}, \\ldots$

\n

$n^\\text{th}$ term = [[0]]

", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "

The formula for the $n^{\\text{th}}$ term of an arithmetic sequence is

\n

\$a_n = a_1 + (n-1)d, \$

\n

where

\n
\n
• $a_n$ is the $n^\\text{th}$ term;
• \n
• $a_1$ is the first term in the sequence;
• \n
• $d$ is the common difference between consecutive terms.
• \n
", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{m[1]*2}", "showFeedbackIcon": true, "prompt": "

For this arithmetic sequence, what is $a_1$?

", "minValue": "{m[1]*2}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{m[1]}", "showFeedbackIcon": true, "prompt": "

What is $d$?

", "minValue": "{m[1]}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "marks": 0}, {"variableReplacements": [], "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answer": "-{m[2]}*n+{m[2]*9+2}", "marks": "3", "variableReplacements": [], "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "

$\\var{m[2]*8+2}, \\var{m[2]*7+2}, \\var{m[2]*6+2}...$

\n

$n^\\text{th}$ term = [[0]]

", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "

The formula for the $n^{\\text{th}}$ term of an arithmetic sequence is

\n

\$a_n = a_1 + (n-1)d, \$

\n

where

\n
\n
• $a_n$ is the $n^\\text{th}$ term;
• \n
• $a_1$ is the first term in the sequence;
• \n
• $d$ is the common difference between consecutive terms.
• \n
", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{m[2]*8+2}", "showFeedbackIcon": true, "prompt": "

For this arithmetic sequence, what is $a_1$?

", "minValue": "{m[2]*8+2}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "{-m[2]}", "showFeedbackIcon": true, "prompt": "

What is $d$?

", "minValue": "{-m[2]}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "marks": 0}], "ungrouped_variables": ["m", "n", "c", "ci", "ni", "b"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Given the first three terms of a sequence, give a formula for the $n^\\text{th}$ term.

\n

In the first sequence, $d$ is positive. In the second sequence, $d$ is negative.

"}, "preamble": {"css": "", "js": ""}, "advice": "

Both of these sequences are linear, or arithmetic, sequences. To find formulas for these sequences we need to identify their first terms and common differences.

\n

#### a)

\n

The formula for the $n^\\text{th}$ term of an arithmetic sequence is

\n

\$a_n = a_1 + (n-1)d \\text{.} \$

\n

$a_1$ is the first term and $d$ the common difference between consecutive terms, which we need to identify.

\n

We can find these by drawing up a table of $a_n$ against $n$, and the differences between consecutive terms.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ 1 2 3 $a_n$ $\\pmb{\\var{m[1]*2}}$ $\\var{m[1]*3}$ $\\var{m[1]*4}$ First differences $\\pmb{\\var{m[1]}}$ $\\pmb{\\var{m[1]}}$
\n

The first term and common difference have been highlighted in bold; we can use these to write the formula for the sequence.

\n

\\begin{align}
a_n &= a_1+(n-1)d \\\\
&= \\var{m[1]*2}+(n-1)\\times\\var{m[1]} \\\\
&= \\var{m[1]}n + \\var{m[1]}\\text{.}
\\end{align}

\n

#### b)

\n

Similar to the part a), we can identify $a_1$ and $d$ for this sequence by drawing a table of $a_n$ against $n$.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ 1 2 3 $a_n$ $\\pmb{\\var{m[2]*8+2}}$ $\\simplify{{m[2]}*7+2}$ $\\simplify{{m[2]}*6+2}$ First differences $\\pmb{\\var{-m[2]}}$ $\\pmb{\\var{-m[2]}}$
\n

The first term and common difference have been highlighted in bold; we can use these to form the formula for the sequence.

\n

\\begin{align}
a_n &=a_1+(n-1)d \\\\
&=\\var{m[2]*8+2}+(n-1)\\times\\var{-m[2]} \\\\
&=-\\var{m[2]}n + \\var{m[2]*9+2}\\text{.}
\\end{align}

"}, {"name": "Compute the partial sum of an arithmetic sequence", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "metadata": {"description": "

Given the first and last terms of a finite arithmetic sequence, calculate the number of elements and then the sum of the sequence.

\n

Each part is broken into steps, with the formula given.

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["m", "first_index", "last_index", "num_terms", "first_term", "last_term", "partial_sum"], "type": "question", "rulesets": {}, "advice": "

We are given the arithmetic sequence

\n

\$\\var{m*first_index}, \\var{m*(first_index+1)}, \\var{m*(first_index+1)}, \\ldots, \\var{last_term} \\text{.} \$

\n

#### a)

\n

Let $n$ be the number of terms in the sequence. Then $a_n = \\var{last_term}$.

\n

To find $n$, we must rearrange the formula for the $n^\\text{th}$ term

\n

\$a_n=a_1+(n-1)d\\text{.}\$

\n

The first term is $a_1 = \\var{first_term}$ and the common difference is $d = \\var{first_term+m} - \\var{first_term} = \\var{m}$.

\n

So we have

\n

\\begin{align}
d&=\\var{m} \\text{,} \\\\
a_1&= \\var{first_term} \\text{,} \\\\
a_n&=\\var{last_term} \\text{.}
\\end{align}

\n

We now substitute these values into the formula for the $n^\\text{th}$ term and rearrange to find $n$.

\n

\\begin{align}
\\var{last_term}&=\\var{first_term}+\\var{m}(n-1) \\\\
\\var{last_term}-\\var{first_term}&=\\var{m}(n-1) \\\\
\\frac{\\var{last_term-first_term}}{\\var{m}}&=(n-1) \\\\
n&=\\var{(last_term-first_term)/m}+1 = \\var{num_terms}\\text{.}
\\end{align}

\n

#### b)

\n

The partial sum of the first $n$ terms of an arithmetic sequence is given by

\n

\$\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n)\\text{.}\$

\n

We know from part a) that

\n

\\begin{align}
n&= \\var{num_terms} \\text{,} \\\\
a_1&= \\var{first_term} \\text{,} \\\\
a_n&= \\var{last_term} \\text{.}
\\end{align}

\n

We substitute these values into the formula, obtaining

\n

\\begin{align}
\\sum\\limits_{i=1}^n{a_i}&=\\frac{n}{2}(a_1+a_n)\\\\
&= \\frac{\\var{num_terms}}{2}(\\var{first_term}+\\var{last_term})\\\\
&= \\simplify[]{ ({num_terms}*({first_term+last_term}))/2} \\\\
&= \\var{partial_sum} \\text{.}
\\end{align}

", "variables": {"last_index": {"description": "

The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.

\n

This is the index of the last term shown.

", "definition": "first_index+num_terms-1", "group": "Ungrouped variables", "name": "last_index", "templateType": "anything"}, "m": {"description": "

Common difference between terms

", "definition": "random(2..10)", "group": "Ungrouped variables", "name": "m", "templateType": "anything"}, "first_index": {"description": "

The sequence shown is a subsequence of the infinite sequence $a_i = m \\times i$.

\n

This is the index of the first term shown.

", "definition": "random(6..14)", "group": "Ungrouped variables", "name": "first_index", "templateType": "anything"}, "num_terms": {"description": "

The number of terms in the sequence.

", "definition": "random(11..30)", "group": "Ungrouped variables", "name": "num_terms", "templateType": "anything"}, "partial_sum": {"description": "

Sum of the terms in the sequence.

", "definition": "num_terms*(first_term+last_term)/2", "group": "Ungrouped variables", "name": "partial_sum", "templateType": "anything"}, "first_term": {"description": "

The first term in the sequence.

", "definition": "m*first_index", "group": "Ungrouped variables", "name": "first_term", "templateType": "anything"}, "last_term": {"description": "

The last term in the sequence.

", "definition": "m*last_index", "group": "Ungrouped variables", "name": "last_term", "templateType": "anything"}}, "statement": "

You are given the following arithmetic sequence:

\n

\$\\var{m*first_index}, \\var{m*(first_index+1)}, \\var{m*(first_index+2)}, \\ldots, \\var{last_term}.\$

", "parts": [{"scripts": {}, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

The formula for the $n^\\text{th}$ term in an arithmetic sequence is

\n

\$a_n=a_1+(n-1)d\\text{,}\$

\n

where

\n
\n
• $d$ is the common difference between consecutive terms;
• \n
• $a_1$ is the first term in the sequence;
• \n
• $a_n$ is the $n^\\text{th}$ term in the sequence.
• \n
"}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "prompt": "

For this arithmetic sequence, what is the value of $a_1$?

", "showFeedbackIcon": true, "allowFractions": false, "minValue": "first_term", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "first_term", "mustBeReduced": false, "marks": "0.2", "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "prompt": "

What is the value of $d$?

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How many terms are there in this sequence?

\n

$n =$ [[0]]

", "stepsPenalty": 0}, {"scripts": {}, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

The sum of an arithmetic sequence $a_1, a_2, \\ldots, a_n$ is calculated using the formula

\n

\$\\sum\\limits_{i=1}^n{a_i}=\\frac{n}{2}(a_1+a_n) \$

\n

where

\n
\n
• $a_1$ is the first term;
• \n
• $a_n$ is the $n^{th}$ term;
• \n
• $n$ is the number of terms.
• \n
"}], "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "partial_sum", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "partial_sum", "mustBeReduced": false, "marks": "2", "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "marks": 0, "prompt": "

Find the sum of the sequence up to and including the term $\\var{last_term}$.

\n

[[0]]

", "stepsPenalty": 0}], "tags": ["arithmetic sequences", "Arithmetic sequences", "Arithmetic Sequences", "common difference", "partial sums", "sequences", "taxonomy", "term number"], "preamble": {"js": "", "css": ""}, "variable_groups": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}}]}], "feedback": {"allowrevealanswer": true, "advicethreshold": 0, "intro": "", "showanswerstate": true, "feedbackmessages": [], "showtotalmark": true, "showactualmark": true}, "timing": {"timeout": {"action": "none", "message": ""}, "allowPause": true, "timedwarning": {"action": "none", "message": ""}}, "showstudentname": true, "percentPass": 0, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": ["random_person"], "custom_part_types": [], "resources": []}