Compute the partial sum of an arithmetic sequence

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From users who are members of Transition to university :

Christian Lawson-Perfect	said	Ready to use	7 years, 8 months ago
Elliott Fletcher	said	Needs to be tested	7 years, 8 months ago
Hannah Aldous	said	Needs to be tested	7 years, 9 months ago

History

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Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

I've tidied up the text, and gone to town on the variables because they were crazy.

Restore this checkpoint

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Elliott Fletcher 7 years, 8 months ago

Published this.

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Elliott Fletcher commented 7 years, 8 months ago

I have rearranged this question quite a bit.

I have made it a two part question, instead of just having one part asking you to find the partial sum of the sequence, as the steps would've been too long if it was just a one-part question and i think a lot of students would find this question difficult without some guidance.

The first part asks the student to calculate the term number of the last term of the sequence and then the second part asks the student to calculate the partial sum.

I have also rewritten the statement and the advice accordingly.

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Needs to be tested

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Has some problems

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Christian Lawson-Perfect 7 years, 9 months ago

Gave some feedback: Has some problems

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Christian Lawson-Perfect 7 years, 9 months ago

Saved a checkpoint:

Fixed a few typos...

You need to show the limits of the sum: $\sum_{i=1}^n$ instead of just $\sum$ .

There are a couple of steps you need to carry out in order to work out the partial sum given a list of the elements:

Identify $a_1$ and $a_n$ .

Calculate $n$ .

These could be in Steps, to guide students through the calculation.

Similarly in part b), you need to identify the first term and common difference; these could be Steps.

You've put i) and ii) in the advice, but not in the question itself! And they're basically the same question, aren't they?

The advice uses $d$ for the common difference, but never introduces it. And if your first term is $a_0$ instead of $a_1$ , the formula for $a_n$ is a lot nicer.

I don't like "We need to find the difference between $20 \rightarrow 30 \rightarrow 40$ ". That's not a sentence! I'd prefer something like "We need to find $d$ , the common difference between consecutive terms in the sequence."

This could be split into two separate questions.

Restore this checkpoint

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Hannah Aldous 7 years, 9 months ago

Created this as a copy of Arithmetic sequences in an ice cream shop.

Name	Status	Author	Last Modified
Arithmetic sequences in an ice cream shop	Ready to use	Hannah Aldous	20/11/2019 14:36
Compute the partial sum of an arithmetic sequence	Ready to use	Hannah Aldous	20/11/2019 14:36
Finding the $n^{\text{th}}$ Term of a Quadratic Sequence	Ready to use	Hannah Aldous	20/11/2019 14:39
Identifying different types of sequences	Should not be used	Hannah Aldous	01/12/2020 16:24
Finding the formula for the $n^{\text{th}}$ term of linear sequences	Ready to use	Hannah Aldous	20/11/2019 14:39
Partial sum of an arithmetic sequence - birthday money	Ready to use	Hannah Aldous	20/11/2019 14:41
Find common difference in arithmetic sequences with gaps	Ready to use	Christian Lawson-Perfect	20/11/2019 14:39
Fill in the gaps in an arithmetic sequence	Ready to use	Christian Lawson-Perfect	20/11/2019 14:39
Write down and apply the formula for an arithmetic sequence.	Ready to use	Christian Lawson-Perfect	20/11/2019 14:46
Finding the Formula for the $n^{\text{th}}$ Term of Linear Sequences	draft	Chris Graham	20/07/2017 15:08
Inbbavathie's copy of Find common difference in arithmetic sequences with gaps	draft	Inbbavathie Ravi	24/07/2017 03:48
Find a particular term of a sequence using the given formula	Ready to use	Christian Lawson-Perfect	20/11/2019 14:38
Johan's copy of Write down and apply the formula for an arithmetic sequence.	Has some problems	Johan Maertens	02/08/2017 17:51
Identifying different types of sequences	draft	Luis Hernandez	31/12/2018 01:38
Calcular la suma parcial de una sucesión aritmética	Ready to use	Luis Hernandez	02/12/2020 15:42
Identificar diferentes tipos de sucesiones...	Ready to use	Luis Hernandez	02/11/2020 02:05
Suma parcial de una secuencia aritmética - Cumpleaños	Ready to use	Luis Hernandez	02/12/2020 15:42
Encontrar la diferencia común en una Secuencia Aritmética	Ready to use	Luis Hernandez	02/12/2020 15:42
Encontrar el término de una sucesión usando la fórmula dada...	Ready to use	Luis Hernandez	02/12/2020 15:42
Encontrar la fórmula para el término $n^{\text {th}}$ de una secuencia lineal	Ready to use	Luis Hernandez	02/12/2020 15:42
Simon's copy of Write down and apply the formula for an arithmetic sequence.	draft	Simon Thomas	25/02/2019 10:20
Simon's copy of Find common difference in arithmetic sequences with gaps	draft	Simon Thomas	25/02/2019 10:24
Simon's copy of Finding the formula for the $n^{\text{th}}$ term of linear sequences	draft	Simon Thomas	25/02/2019 10:30
Identifying different types of sequences	draft	Xiaodan Leng	10/07/2019 21:48
Write down and apply the formula for an arithmetic sequence.	draft	Xiaodan Leng	11/07/2019 01:48
Fill in the gaps in an arithmetic sequence	draft	Xiaodan Leng	11/07/2019 01:48
Find common difference in arithmetic sequences with gaps	draft	Xiaodan Leng	11/07/2019 01:49
Compute the partial sum of an arithmetic sequence	draft	Xiaodan Leng	11/07/2019 01:51
Partial sum of an arithmetic sequence - birthday money	draft	Xiaodan Leng	11/07/2019 01:54
Arithmetic sequences in an ice cream shop	draft	Xiaodan Leng	11/07/2019 01:55
Finding the formula for the $n^{\text{th}}$ term of linear sequences	draft	Xiaodan Leng	11/07/2019 01:55
Find a particular term of a sequence using the given formula	draft	Xiaodan Leng	11/07/2019 01:56
Finding the $n^{\text{th}}$ Term of a Quadratic Sequence	draft	Xiaodan Leng	11/07/2019 01:57
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

You are given the following arithmetic sequence:

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

Give any introductory information the student needs.

			Name	Type	Generated Value

m	integer	8
first_index	integer	7
last_index	integer	21
num_terms	integer	15
first_term	integer	56
last_term	integer	168
partial_sum	rational	1680

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..10)

JME function reference

Description

Common difference between terms

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

Can an exam override the value of this variable?

Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Number entry
  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 many terms are there in this sequence?

$n =$ Unnamed gap

Advice

We are given the arithmetic sequence

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

a)

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

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

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

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

So we have

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

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

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

b)

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

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

We know from part a) that

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

We substitute these values into the formula, obtaining

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

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This question is used in the following exams:

Sequences by Elliott Fletcher in Elliott's workspace.
Arithmetic sequences by Christian Lawson-Perfect in Transition to university.
Arithmetic sequences by steve kilgallon in Graphs and series.
P12 - Series & Sequences by Rob Beckett in Rob's workspace.
Arithmetic sequences by Simon Thomas in Maths support.
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Jean jinhua's copy of Arithmetic sequences by Jinhua Mathias in Maths 3-2020.
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claire's copy of Arithmetic sequences by claire lines in claire's workspace.
Wenling's copy of Arithmetic sequences by Wenling Zhang in Wenling's workspace.
Engineering First Year Skills Audit by S Russell in Engineering Maths booster.
First Year Undergraduate Computer Science Skills Audit by S Russell in Computer Science Maths Booster.

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Compute the partial sum of an arithmetic sequence

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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

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Hannah Aldous 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

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Compute the partial sum of an arithmetic sequence

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

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Hannah Aldous 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`