// Numbas version: exam_results_page_options {"name": "Partial sum of an arithmetic sequence - birthday money", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["person", "pronouns"], "name": "A person"}], "extensions": ["random_person"], "name": "Partial sum of an arithmetic sequence - birthday money", "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.

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

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "advice": "

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.

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

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

a)

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

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

where

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

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

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

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

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

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

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

b)

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']} 21^st birthday.

This question involves calculating the sum using the equation

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

We know from part a) that

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

Using our formula for the sum,

\\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}

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:

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

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

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A random person

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first term in the sequence

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

£[[0]].

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Use the arithmetic formula,

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

where

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

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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 $a_{21}$.

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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?

£[[0]].

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

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

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