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Infinite Surds. As we can see there are ten terms of this sequence where is the general term of the sequence when, is the first term of the sequence.

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Math Portfolio

Infinite Surds


a surd is an irrational number that can not be written as a fraction of two integers but can only be expressed using the root sign.

Bellow an example of an infinite surd:


This surd can be turned into a set of particular numbers sequence:












As we can see there are ten terms of this sequence where image09.pngis the general term of the sequence whenimage10.png,image11.png is the first term of the sequence...Etc.

A formula has been defined for image12.pngin terms ofimage09.png:



...read more.


 gets closer to a fixed value.

To investigate more about this fixed value we take this equation image18.png into consideration as n gets bigger.




- 0.13956

















We can figure out from the table above that when n gets larger, the term (an+an+1) gets closer to zero but it never reaches it

So we can come to the conclusion:

When n approaches infinity, lim (an-an+1) →0

An expression can be obtained in the case of the relation between n and an to get the exact value of the infinite surd:

image19.png= x

If we apply formula (1) to this:


x2=1+x   →   image22.png

The equation can be solved using the solution of a quadric equation:


Whereas a=1, b=-1, c=-1

Two solutions for x were obtained:

x=1.618033989 and

...read more.



The negative solution is ignored so:


To find some values of k to make the expression an integer:

We can see that 4k is an even number and 4k+1 is odd, so image46.png is an odd number if 4k+1 is a perfect square hence 1+image46.png is an even number and possible to be divided by 2. As a result if 4k+1 is a perfect square we can obtain an integral number in the result.

For example let k=2:



image48.png← not integer because13 is not a perfect square

Thus we come to the conclusion that only limited values of k can be used to make the result an integer and those values are any value of k can make 4k+1 a perfect square such as k = 2,6,12…etc.

...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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