Simon's copy of Finding the formula for the $n^{\text{th}}$ term of linear sequences

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Simon Thomas 6 years, 1 month ago

Created this as a copy of Finding the formula for the

$n^{\text{th}}$ term of linear sequences.

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

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Question statement

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

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:

Give any introductory information the student needs.

			Name	Type	Generated Value

m	list	[ 5, 7, 10, 5, 2 ]
n	list	[ 3, 4, 2, 2, 2, 1, 4 ]
c	list	[ 7, 5, 13, 9, 13, 9, 13, 5 ]
ci	list	List of 10 items
ni	list	List of 10 items
b	list	[ 3, 3, 2, 4, 3 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
repeat(random(2..10),5)

JME function reference

Description

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

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Generated value: `list`

[ 5, 7, 10, 5, 2 ]

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

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

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

$n^\text{th}$ term = Unnamed gap

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.

a)

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

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

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

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

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

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

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

b)

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

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

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

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

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This question is used in the following exam:

Arithmetic and Geometric Series by Simon Thomas in Maths support.

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Simon's copy of Finding the formula for the $n^{\text{th}}$ term of linear sequences

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Simon Thomas 6 years, 1 month ago

Error in variable testing condition

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Generated value: `list`

Parts

Gap-fill

a)

b)

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Simon's copy of Finding the formula for the nthn^{\text{th}} term of linear sequences

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 1 month ago

Error in variable testing condition

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Warning

Generated value: list

Parts

Gap-fill

a)

b)

Simon's copy of Finding the formula for the $n^{\text{th}}$ term of linear sequences

Generated value: `list`