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

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Xiaodan Leng 5 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

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
c	integer	3
b	integer	3
sequence	list	[ 3, 9, 21, 39, 63, 93 ]
first_difference	list	[ 6, 12, 18, 24, 30, 36 ]

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Generated value: `integer`

→ Used by:

first_difference
sequence

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

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Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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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

History

Comment

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Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

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Generated value: integer

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

Generated value: `integer`