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Mathematics portfolio on Infinite Surd

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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.


Consider this surd as a sequence of terms an where:











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

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




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.











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

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


The value of an infinite surd is not always an integer






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,


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






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



...read more.

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