Find point when a sequence becomes strictly decreasing

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Bill Foster

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Bill Foster 6 years, 3 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

Let $\{x_n\}$ be the sequence given by

$x_n=n^\var{k} \var{t}^n$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	number	0.9765021474
b	number	0.0234978526
c	number	41.5570802897
n	integer	42
t	number	0.7
k	integer	15

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Name

Data type

Value

xxxxxxxxxx
 
t^(1/k)

JME function reference

Description

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Generated value: `number`

0.9765021474

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Parts

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1. Gap Number entry
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Find the smallest integer $N$ such that $x_{m+1} \leq x_m$ for all $m \geq N$ .

The smallest integer is Unnamed gap

Advice

The condition $x_{m+1} \leq x_m$ is $(m+1)^\var{k} \var{t}^{m+1} \leq m^\var{k} \var{t}^m$ . This can be shown to be equivalent to $m \geq \dfrac{\var{t}^{1/\var{k}}}{1-\var{t}^{1/\var{k}}}=\var{c}$ . We take $N$ to be the smallest integer $\geq \var{c}$ , so $N=\var{n}$ . Then $m \geq \var{c}$ for all $m \geq \var{n}$ and therefore $x_{m+1} \leq x_m$ for all $m \geq \var{n}$ .

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Part	Test	Passed?
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Find point when a sequence becomes strictly decreasing

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Taxonomy: mathcentre

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Bill Foster 6 years, 3 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `number`

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Gap-fill

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Error

Find point when a sequence becomes strictly decreasing

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 3 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`