- Level: International Baccalaureate
- Subject: Maths
- Word count: 1579
Math Portfolio type 1 infinite surd
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Introduction
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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:
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
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 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:
However, when it comes to exact value:
Using quadratic formula to solve this equation:
Notice: This value is canceled out because the value of a square root can not be negative.
Thus:
Consider the general infinite surd:
The first term is:
The an+1term in term of an:
An expression for the exact value of this general infinite surd in terms of k:
Using quadratic formula to solve this equation:
Thus:
The exact value of this general infinite surd in terms of k is:
As we can see the value of an infinite surd is not always an integer.
Example:
Conclusion
After checking the values table of the infinite surd and . 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
This infinite surd is
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 and this infinite surds in particular and this 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’/
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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