Partial sum of an arithmetic sequence - birthday money

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Christian Lawson-Perfect	said	Ready to use	7 years, 8 months ago
Elliott Fletcher	said	Needs to be tested	7 years, 8 months ago
Chris Graham	said	Has some problems	7 years, 8 months ago
Hannah Aldous	said	Needs to be tested	7 years, 8 months ago

History

○

Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Ready to use

○

Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

Lots of little edits - I basically disagreed with every formatting choice you made!

This is a good question.

Restore this checkpoint

○

Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Needs to be tested

○

Chris Graham 7 years, 8 months ago

Gave some feedback: Has some problems

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Chris Graham commented 7 years, 8 months ago

Ok, so we'e basically torn up and re-written the original question... This looks good, however as for previous questions on arithmetic series, can we have a table of the values and differences in the advice to part (a) please.

I would change "by his" in the statement to "after his", as it implies the amount before he receives money on his birthday. I know that you state that it includes the amount after, but this appears contradictory.

I'm not keen on $n$ being "the amount of numbers in a sequence". So if you have 10 terms in the sequence but you're calculating the $a_5$ term, do you still use $n=10$ ? "Term number" is terminology that is often used.

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Chris Graham commented 7 years, 8 months ago

Sorry, you are quite right, I meant to say interest! If you are wording the question in a way that it is "What contribution...?" then you don't really need this anyway, so I have removed those references.

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Elliott Fletcher 7 years, 8 months ago

Published this.

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Needs to be tested

○

Elliott Fletcher commented 7 years, 8 months ago

Is inflation the right term that we need or should it be interest?

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Chris Graham commented 7 years, 8 months ago

As discussed, the sum of candles on a birthday cake over several years makes little sense either. Why would you want to know?

Parents adding money to a bank account is much better, and then you could ask (excluding inflation) how much money the parents have added over $n$ years.

This can also be split into two parts, firstly calculate $a_n$ for the given $n$ , and then the sum of the series, with a step for each part.

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Chris Graham commented 7 years, 8 months ago

The step doesn't give enough useful information, and neither does it give the opportunity for the student to break down the question: so the first step would be to find $a_n$ , but the step doesn't tell me how to do that. There's an easy opportunity to have gaps here for the common difference, the final term, and so on.

I also have major issues with the context. I'm led to believe Bruce receives 93 slices of his birthday cake on his 50th birthday. How big are these slices? I can only assume that they are getting smaller at a rate inversely proportional to the increase in slices. How about changing to the number of candles? 93 candles on a cake is still a stretch, but slightly more plausible.

○

Chris Graham 7 years, 8 months ago

Gave some feedback: Has some problems

○

Hannah Aldous 7 years, 8 months ago

Gave some feedback: Needs to be tested

○

Hannah Aldous 7 years, 8 months ago

Created this as a copy of Compute the partial sum of an arithmetic sequence.

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Ugur's copy of Write down and apply the formula for an arithmetic sequence.	Ready to use	Ugur Efem	12/01/2022 11:45

There are 47 other versions that do you not have access to.

Question 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.

Give any introductory information the student needs.

			Name	Type	Generated Value

			person	dict	Dictionary with 3 entries
			pronouns	dict	Dictionary with 5 entries

			Name	Type	Generated Value

m	list	[ 5, 8, 3, 10, 8 ]
n	list	[ 4, 7, 5, 5, 5, 8, 3 ]
c	list	[ 4, 3, 7, 8, 7, 7, 5, 6 ]
ci	list	List of 10 items
ni	list	List of 10 items
b	list	[ 15, 25, 10 ]
first	integer	15

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random_person()

JME function reference

Description

A random person

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `dict`

[ "gender": safe("male"), "name": safe("James"), "pronouns": [ "they": safe("he"), "their": safe("his"), "theirs": safe("his"), "them": safe("him"), "themself": safe("himself") ] ]

→ Used by:

pronouns

Statement
Advice
"Part a)" - prompt
"Part b)" - prompt

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) Gap-fill
1. Gap Number entry
  Add
1. Step Information only

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

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?

£Unnamed gap.

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.

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

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

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Part	Test	Passed?
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Number entry
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Number entry
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Gap-fill
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Number entry
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Partial sum of an arithmetic sequence - birthday money

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `dict`

Parts

Gap-fill

a)

b)

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Partial sum of an arithmetic sequence - birthday money

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: dict

Parts

Gap-fill

a)

b)

Generated value: `dict`