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

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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
Chris Graham	said	Has some problems	7 years, 8 months ago
Bradley Bush	said	Needs to be tested	7 years, 9 months ago
Hannah Aldous	said	Needs to be tested	7 years, 9 months ago
Vicky Hall	said	Has some problems	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 reworded a bit.

Restore this checkpoint

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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

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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

Same issue here with "The amount of numbers in a sequence". Can you re-write anywhere else that we have this phrase please?

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Elliott Fletcher commented 7 years, 8 months ago

I've added steps to both parts

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Needs to be tested

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Chris Graham commented 7 years, 8 months ago

Elliott, can you add a step to both parts please, giving the formula and asking the student to identify $a_1$ and $d$

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Chris Graham 7 years, 8 months ago

Gave some feedback: Has some problems

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Bradley Bush 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Bradley Bush commented 7 years, 9 months ago

Thank you for the feedback Christian,

There was a slight error with the gap fills so I fixed that

There was another error with the answer marking, so that's fixed too

I've adjusted the advice as you recommended to include a table, but I've also left Hannah'smethod as an alternative, I'm not sure whether yu wanted it completely replaced or no so I've left it in.

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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:

I've split this into two parts - why make up your own part headers when Numbas can do it for you? It's important to let the student submit their answer to one part before moving on to another.

This question could very easily have steps - identify the first term, identify the common difference, and then you can write down the formula.

I've added a sentence to the top of the advice describing what needs to be done. A "plan of action" like this makes it easier to follow the more detailed parts of the solution.

I'd work this out by drawing up a table:

$n$ 1 2 3

$a_n$ 18 27 36

First differences 9 9

The formula is $a_n = a_1 + (n-1)d$ . I can read off from the table that $a_1 = 18$ and $d = 9$ .

There's a reason you're taught to do this stuff using tables at school - visually, you can see how $n$ , $a_n$ , and the common difference relate to each other as the sequence progresses. Without this, it can all seem like aimlessly shuffling numbers about.

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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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Has some problems

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Elliott Fletcher commented 7 years, 9 months ago

Main Parts

I think you should label each of the questions i) and ii) as you have done in the advice section

I think the expected answer in both of these questions is wrong, the answer that you have in the advice section is right, i think its just that you calculated the adding term using n=0 for the displayed answer.

Advice

i) where you have written nth term here it doesn't really display correctly, it looks like n^(t)h instead of n^th.

I think you should say "Our nth term will take the form 8n+b, where b is some constant"

ii) i think "them" should be "it"

I think you should say "Our nth term will take the form 5n+c, where c is some constant"

I think you use the wrong number when you say how much the original sequence is greater than the sequence for 5n by.

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Vicky Hall 7 years, 9 months ago

Gave some feedback: Has some problems

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Vicky Hall commented 7 years, 9 months ago

Change the question to 'find the formula for the $n$ th term'. It would be good to have one increasing and one decrasing sequence to show that sometimes the $n$ coefficient is negative.

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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 Finding the

$n^{\text{th}}$ Term of a Quadratic Sequence.

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

A linear 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, 8, 6, 6, 8 ]
n	list	[ 1, 4, 2, 1, 2, 4, 1 ]
c	list	[ 6, 11, 12, 6, 3, 4, 4, 6 ]
ci	list	List of 10 items
ni	list	List of 10 items
b	list	[ 3, 4, 4, 2, 3 ]

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Name

Data type

Value

xxxxxxxxxx
 
repeat(random(2..10),5)

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Description

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

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Generated value: `list`

[ 5, 8, 6, 6, 8 ]

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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 exams:

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Finding the formula for the nthn^{\text{th}} term of linear sequences

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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 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Bradley Bush 7 years, 9 months ago

Bradley Bush commented 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

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Vicky Hall 7 years, 9 months ago

Vicky Hall commented 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Error in variable testing condition

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Warning

Generated value: list

Parts

Gap-fill

a)

b)

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

Generated value: `list`