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Math Portfolio type 1 infinite surd

Extracts from this document...

Introduction

THIS DOCUMENT WAS DOWNLOADED FROM COURSEWORK.INFO - THE UK'S COURSEWORK DATABASE - HTTP://WWW.COURSEWORK.INFO/

TRAN NGUYEN ANH LINH, 5Y

MATH PORTFOLIO

SL Portfolio Type 1

(Mathematical Investigation)

Infinite Surds

2009-2010

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:

image00.png

And I will use data table and graph to suggest about the value of image01.png 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:

image06.png

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

image12.png

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

...read more.

Middle

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


Evaluation:

(I repeat the entire process above)

As the graph shown, as the n gets very large, the values of an still be the same. Hence the value of image01.png is equal to 0. According to the data table, the a8, a9 and a10 already have the same value 1.99999, equivalent until 5 decimal place.And after 15th term, all the value are exactly the same values 2.

As we have:

image09.png

However, when it comes to exact value:

image10.png

Using quadratic formula to solve this equation:

image04.png

Notice:image05.png This value is canceled out because the value of a square root can not be negative.

Thus:

image11.png


Consider the general infinite surd:

image12.png

The first term is: image13.png

The an+1term in term of an:

image14.png

An expression for the exact value of this general infinite surd in terms of k:

image15.png

Using quadratic formula to solve this equation:

image04.png

Thus:

image16.png

The exact value of this general infinite surd in terms of k is:

image17.png


As we can see the value of an infinite surd image12.png is not always an integer.

Example:

image18.png

...read more.

Conclusion

=SQRT(6+SQRT(6)) then following to the next values will be B3=SQRT(6+B2) etc…

After checking the values table of the infinite surd image32.png and image34.png. Hence, the general statement that represents all the values of k for which the expression is an integer is correct.

Now I will test the 2nd method using a greater number, for example 99

If

image35.png

This infinite surd is image36.png

Table shows the first ten terms of this infinite surd:

(Using MS Excel 2008)

Terms

Values

1

99.99749369

2

99.99998747

3

99.99999994

4

100

5

100

6

100

7

100

8

100

9

100

10

100

11

100

12

100

13

100

14

100

15

100

16

100

17

100

18

100

19

100

20

100

21

100

22

100

23

100

24

100

25

100

Notice:

The formula to calculate the second column: B2=SQRT(9900+SQRT(9900)) then following to the next values will be B3=SQRT(9900+B2) etc…

Therefore, the 2nd method also correct as which the exact value is 100, an even and divided-able by 2 integer.


III. Conclusion:

After all the tests and checking the general statement, they prove that my result are true and correct for all the values of k. Nonetheless, I have also solved the other questions at the beginning of this portfolio and found the general formula for the exact value of this image00.png and this image06.png infinite surds in particular and this image12.png surd formula in general with difference method and examples. As this part, I have done all the question of the mathematics portfolio for IB students ‘Infinite Surd’/

...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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