Full binomial expansion

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Simon Thomas 6 years, 1 month ago

Created this as a copy of First 3 terms of Binomial series for Natural exponent.

Name	Status	Author	Last Modified
Binomial series for Natural exponent	Ready to use	Frank Doheny	20/03/2018 08:25
Binomial series for non-Natural exponent	Ready to use	Frank Doheny	20/03/2018 08:25
First 3 terms of Binomial series for Natural exponent	Ready to use	Frank Doheny	20/03/2018 08:24
Gemma's copy of Binomial series for Natural exponent	draft	Gemma Crowe	14/08/2018 15:39
Gemma's copy of First 3 terms of Binomial series for Natural exponent	draft	Gemma Crowe	14/08/2018 13:42
Algebra: binomial expansion, coefficient of particular term, calculator	Ready to use	Lovkush Agarwal	06/07/2020 11:30
Full binomial expansion	draft	Simon Thomas	27/02/2019 09:40
Simon's copy of First 3 terms of Binomial series for Natural exponent	draft	Simon Thomas	27/02/2019 10:23
Simon's copy of Binomial series for Natural exponent	draft	Simon Thomas	27/02/2019 09:44
Simon's copy of Binomial series for non-Natural exponent	draft	Simon Thomas	27/02/2019 10:24

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Question statement

Given the expression $\simplify{({a}+{b}x)^{n}}$

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			Name	Type	Generated Value

a	integer	2
b	integer	-1
n	integer	5
c	number	-40
k	integer	3
ellipsis	string	... +

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Generated value: `integer`

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The binomial series expansion for an expression of the form $(a+bx)^n$ where $n$ is a Natural number is given by:

$(a+bx)^n=\tbinom{n}{0}(a)^n(bx)^{0}+\tbinom{n}{1}(a)^{n-1}(bx)^{1}+\tbinom{n}{2}(a)^{n-2}(bx)^{2}+...+\tbinom{n}{k}(a)^{n-k}(bx)^{k}+...+\tbinom{n}{n}(a)^{0}(bx)^{n}$

In this example $n=\var{n}$ , $a=\var{a}$ and $b=\var{b}$ .

So the binomial series expansion is:

$\var{a}^{\var{n}}+\tbinom{\var{n}}{\var{1}}\times\var{a}^{\var{n}-1}\times(\var{b}x)^{1}+\tbinom{\var{n}}{2}\times\var{a}^{\var{n}-2}\times(\var{b}x)^{2}+\var{ellipsis}\tbinom{\var{n}}{\var{n}}\times\var{a}^{\var{n}-\var{n}}\times(\var{b}x)^\var{n}$

$=\simplify{{a}^{n}+{n}*{a}^({n}-1)*{b}x+{n}*{n-1}*{a}^{{n}-2}*{{b}^2}/2x^2+{n}*{n-1}*{n-2}*{a}^{{n}-3}*{{b}^3}/6x^3+{n}*{n-1}*{n-2}*{n-3}*{a}^{{n}-4}*{{b}^4}/24x^4+{n}*{n-1}*{n-2}*{n-3}*{n-4}*{a}^{{n}-5}*{{b}^5}/120x^5}$

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This question is used in the following exam:

Binomial expansion quiz by Simon Thomas in Algebra.

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Simon Thomas 6 years, 1 month ago

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Full binomial expansion

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

Error

Warning

Generated value: integer

Parts

Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

Pattern restriction

Variables

String restrictions

Generated value: `integer`