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Mathematics portfolio on Infinite Surd

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Introduction

Mathematics Portfolio

Standard Level Type I

Infinite Surds

Germaine An

A surd is a sum with one or more irrational number expressed with a radical sign as addends. Examples are 1+√3, √2+√3, and √(1+√(1+√1)). Therefore, an infinite surd has an infinite number of such addends. An example is in the diagram.


The following expression is an example of an infinite surd.

image00.png

Consider this surd as a sequence of terms an where:

image01.pngimage01.png=image50.pngimage50.png=1.414213562

image61.pngimage61.png=image69.pngimage69.png=1.553773974

image02.pngimage02.png=image08.pngimage08.png=1.598053182

image17.pngimage17.png=image28.pngimage28.png=1.611847754

image34.pngimage34.png=image44.pngimage44.png=1.616121207

image45.pngimage45.png=image46.pngimage46.png=1.617442799

image47.pngimage47.png=image48.pngimage48.png=1.617851291

image49.pngimage49.png=image51.pngimage51.png=1.617977531

image52.pngimage52.png=image53.pngimage53.png=1.618016542

image54.pngimage54.png=image55.pngimage55.png=1.618028597

According to the result, you can aware that a2 = image56.pngimage56.png

Then analyzing the formula, an+1 = image57.pngimage57.png

image58.png

On the graph, it represents that at the point of the beginning it raises rapidly as acceleration. However, after that, an-an+1

...read more.

Middle

So,

1.618033989

Consider another infinite surd

image66.pngimage66.png where the first term is image67.pngimage67.png

Repeat the entire process above to find the exact value for this surd.

image68.pngimage68.png=image67.pngimage67.png=1.847759065

image06.pngimage06.png=image70.pngimage70.png=1.961570561

image71.pngimage71.png=image72.pngimage72.png=1.990369453

image73.pngimage73.png=image74.pngimage74.png=1.997590912

image75.pngimage75.png=image76.pngimage76.png=.1999397637

image77.pngimage77.png=image78.png=1.999849404

image79.pngimage79.png=image80.pngimage80.png=1.999962351

image81.pngimage81.png=image82.pngimage82.png=1.999990588

image83.pngimage83.png=image03.pngimage03.png=1.999997647

image04.pngimage04.png=image05.pngimage05.png=1.999999412

According to the result, you can aware that image06.pngimage06.png =image07.pngimage07.png

Then analyzing the formula, image09.pngimage09.png=image10.pngimage10.png

image11.png

This graph shows us that there is a huge change between 1 and 2. And it is constant from 4.  . bn and image12.pngimage12.png-image09.pngimage09.png

...read more.

Conclusion

The value of an infinite surd is not always an integer

2=1x2

6=2x3

12=3x4

20=4x5

30=5x6

These are proved, using this formula k= n (n+1)

5(5+1) = 30

5th term which is also described as 5x6=30

Test the validity of your general statement using other values of k.

First off, the formula is k= n (n+1)

Apply image13.pngimage13.png

With using quadratic equation,

image37.png

8x9=72----y = image38.pngimage38.png

=46

88x89=7832------image39.pngimage39.png

=89

222x223=49506---------image40.pngimage40.png

=223

And so on..

It is able to notice that

When k is 72, y = image41.pngimage41.png =46

Discuss the scope and/or limitations of your general statement.

According to this, we’ve found that value of K and n always are

image42.png

nimage43.pngimage43.png

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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