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

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Christian Lawson-Perfect	said	Ready to use	7 years, 8 months ago
Hannah Aldous	said	Needs to be tested	7 years, 9 months ago
Lauren Richards	said	Has some problems	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 vastly rewritten the advice, and given the variables sensible names.

It would be good to have another version of this question where you fill in the table first.

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Elliott Fletcher 7 years, 8 months ago

Published this.

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Aiden McCall 7 years, 8 months ago

Saved a checkpoint:

I have edited the advice. I think the spacing/positioning could be done better but am not sure how to do this with the tables so they don't stretch when you move each part.

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Aiden McCall 7 years, 9 months ago

Saved a checkpoint:

The advice in this question is very hard to follow. I have an alternative method and am going to change the advice. In case my method is not preferred or wants to be reverted I have made a checkpoint here.

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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

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 commented 7 years, 9 months ago

Also, note that I used "first differences" and "second differences" - I thnk Lauren also spotted that it's tricky to keep track of all the differences floating about.

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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:

The layout of the advice is very hard to follow.

I would work this out by drawing up a table with rows for $n$ , $a_n$ , and first and second differences.

For example,

$n$ 1 2 3 4

$a_n$ 15 22 33 48

First differences 7 11 15

Second differences 4 4

Because the second difference is $4$ , I know that the coefficient of $x^2$ is $4/2 = 2$ .

Then I can add a row for $a_n - 2n^2 = bn+c$ :

$n$ 1 2 3 4

$a_n$ 15 22 33 48

$2n^2$ 2 8 18 32

$a_n-2n^2$ 13 14 15 16

So $bn+c = n+12$ , and the formula for the whole sequence is $a_n = 2n^2+n+12$ .

Sometimes, a good layout can save a lot of verbiage!

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Lauren Richards 7 years, 9 months ago

Gave some feedback: Has some problems

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Lauren Richards commented 7 years, 9 months ago

So after you have found the difference between the sequential terms, and then found the difference between the differences, which you found in the question I did to be 8, it is not clear how you would decide to start the nth term off as 8/2n^2. I think you need to add in a line explaining your reasoning for this.

I don't think the first sentence reads very well. Maybe something more like "We first need to calculate the differences between the terms in the sequence.". and then for the second bit of writing "We can then calculate the differences of the term differences." I totally get that that is quite hard to word without being confusing.

I think "original sequence" should probably be capitalised.

In the advice, the second bit of writing that finishes with the word differences, is missing a full stop. You're also missing a comma at the end of "difference" near the end of the advice but I think this sentence should be re-worded anyway as it starts with "If", but then doesn't really end it like you would expect. I think it should be something like "We should first find the common difference.".

No fault of yours but it gets a little confusing with the amount of differences being mentioned, and I think when you refer to the "sequence of differences" you should have sequence of differences in brackets when you first make it to make it clear when you next refer to it.

I think differences sequence should be capitalised.

At the end, you're calculating the difference between 2n and the sequence of differences but you haven't done that in the same way as you have done earlier in the question advice and I think you probably should do.

One time I did the question, it said "Then we can find the difference between 1n and our sequence of differences" so I think you should make sure it just states "n" in this case.

I think this is a good question, and tricky to make clear!

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

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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

I would reword the question to say 'Find the formula for the $n$ th term of the quadratic sequence'.

The advice is good but think the last part, where you find the constant, would look better if it were laid out like the previous part, so you show the student what the sequence $3n^2+3n$ looks like compared to the original sequence. You also need to remove the reference to part b).

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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 Compute the partial sum of an arithmetic 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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
c	integer	4
b	integer	1
sequence	list	[ 4, 8, 18, 34, 56, 84 ]
first_difference	list	[ 4, 10, 16, 22, 28, 34 ]

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Generated value: `integer`

→ Used by:

first_difference
sequence

Advice
"Unnamed part" - prompt
"Unnamed part" → "Unnamed gap" - Correct answer

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1. Gap Mathematical expression
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Find the formula for the $n^\text{th}$ term of the quadratic sequence

$\var{a*1^2+b*1+c}, \var{a*2^2+b*2+c}, \var{a*3^2+b*3+c}, \var{a*4^2+b*4+c}, \ldots$

$a_n =$ Unnamed gap

Advice

A quadratic sequence is a sequence in which the $n^\text{th}$ term takes the form $an^2+bn+c$ .

Unlike an arithmetic sequence, a quadratic sequence does not have a constant common difference, but instead has differences which increase or decrease by a constant value. That is, the second differences are constant.

An easy way of determining the constants $a$ , $b$ and $c$ is to draw up a table of $n$ , $a_n$ , and the first and second differences between terms.

$n$		$1$	$2$	$3$	$4$
$a_n$	Algebraic form	$a+b+c$	$4a+2b+c$	$9a+3b+c$	$16a+4b+c$
	Numerical form	$\var{sequence[1]}$	$\var{sequence[2]}$	$\var{sequence[3]}$	$\var{sequence[4]}$
First difference	Algebraic form		$3a+b$	$5a+b$	$7a+b$
	Numerical form		$\var{first_difference[1]}$	$\var{first_difference[2]}$	$\var{first_difference[3]}$
Second difference	Algebraic form			$2a$	$2a$
	Numerical form			$\var{2a}$	$\var{2a}$

In this table, the algebraic forms for $a_n$ have been found by substituting $n$ into the formula. The first and second differences are found by subtracting adjacent entries in the row above.

We can see straight away that the second difference is always $2a = \var{2a}$ , so $a = \var{a}$ .

Next, we can substitute the value of $a$ into the algebraic form of one of the first differences, and rearrange to find $b$ .

$\begin{align} 3 \times \var{a} + b &= \var{3*a+b} \\ \var{3a} + b &= \var{3a+b} \\ b &= \var{b} \text{.} \end{align}$

Finally, substitute $a$ and $b$ into the algebraic form for any of the $a_n$ ( $a_1$ is simplest) to find $c$ .

$\begin{align} \var{a} + \var{b} + c &= \var{a+b+c} \\ c &= \var{c} \text{.} \end{align}$

Now we can write down the general formula for $a_n$ .

$a_n = \simplify{{a}n^2+{b}n+{c}} \text{.}$

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This question is used in the following exams:

Sequences by Elliott Fletcher in Elliott's workspace.
Transition Frequencies by Chris Templet in Chris's workspace.

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

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Taxonomy: mathcentre

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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

Aiden McCall 7 years, 8 months ago

Aiden McCall 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Christian Lawson-Perfect 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

Lauren Richards 7 years, 9 months ago

Lauren Richards commented 7 years, 9 months ago

Hannah Aldous 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: `integer`

Parts

Gap-fill

$n$	1	2	3	4
$a_n$	15	22	33	48
First differences		7	11	15
Second differences			4	4

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Finding the nthn^{\text{th}} Term of a Quadratic Sequence

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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

Aiden McCall 7 years, 8 months ago

Aiden McCall 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Christian Lawson-Perfect 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

Lauren Richards 7 years, 9 months ago

Lauren Richards commented 7 years, 9 months ago

Hannah Aldous 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

Error

Warning

Generated value: integer

Parts

Gap-fill

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

Generated value: `integer`