Math Portfolio type 1 infinite surd

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SL Portfolio Type 1

(Mathematical Investigation)

Infinite Surds


Launch Date: 7 Oct 2009 (Wed)

Submission Deadline: 21 Oct 2009 (Wed), 3 pm.

This great work

is done by

Anh Linh, 5Y

I. Introduction:

In this portfolio, firstly I will find out the equation for the nth term of this infinite surd:

And I will use data table and graph to suggest about the value of  when it gets very large. After that I will prove an equation to calculate the exact value of the infinite surd. Followed by repeating all the same steps above but for the other infinite surd:

After I repeated all the steps for this surd, I will consider about the general infinite surd:


Using all the steps that I did for the others two surd, I will find an expression for the exact value of this general infinite surd in terms of k. Furthermore, I will find the general statement that represents all the values of k for which the expression is an integer.

Nonetheless, I will discuss the scope and limitations of my general statement. And finally I will explain how I arrived my general statement and the integer expression.


The following expression is an example of an infinite surd:


Consider this surd as a sequence of term an where:




The formula for an+1 in terms of an:


Calculate the decimal values of the first ten terms of the sequence:

Using MS Excel 2008

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The formula to calculate the second column: B2=SQRT(1+SQRT(1)) then following to the next values will be B3=SQRT(1+B2) etc…

Plot the relation graph between n and an:


(This graph I used the software Microsoft Excel 2008 to draw)


As the graph shown, as the n gets very large, the values of an still be the same. Hence the value of  is equal to 0. According to the data table, after the 4th term, all the data have the same until 1.61 which is 2 decimal place. And after a18, all the values are exactly have the same ...

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