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:

==1.414213562

==1.553773974

==1.598053182

==1.611847754

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==1.616121207

==1.617442799

==1.617851291

==1.617977531

==1.618016542

==1.618028597

According to the result, you can aware that a2 =  

Then analyzing the formula, an+1 =  

On the graph, it represents that at the point of the beginning it raises rapidly as acceleration. However, after that, an-an+1 value has been had no huge change which means that difference is close to 0.

Apply

 

Here is a proved formula

a=

a2=1+a

a2-a-1=0

Use quadratic equation

   

 a= = 1.618033989 or -0.6180339887

However, the root cannot be ...

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