- Level: International Baccalaureate
- Subject: Maths
- Word count: 546
Mathematics portfolio on Infinite Surd
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Introduction
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
==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
Middle
So,
∴ 1.618033989
Consider another infinite surd
where the first term is
Repeat the entire process above to find the exact value for this surd.
==1.847759065
==1.961570561
==1.990369453
==1.997590912
==.1999397637
==1.999849404
==1.999962351
==1.999990588
==1.999997647
==1.999999412
According to the result, you can aware that =
Then analyzing the formula, =
This graph shows us that there is a huge change between 1 and 2. And it is constant from 4. . bn and -
Conclusion
The value of an infinite surd is not always an integer
2=1x2
6=2x3
12=3x4
20=4x5
30=5x6
These are proved, using this formula k= n (n+1)
5(5+1) = 30
5th term which is also described as 5x6=30
Test the validity of your general statement using other values of k.
First off, the formula is k= n (n+1)
Apply
With using quadratic equation,
8x9=72----→y =
=46
88x89=7832------→
=89
222x223=49506---------→
=223
And so on..
It is able to notice that
When k is 72, y = =46
Discuss the scope and/or limitations of your general statement.
According to this, we’ve found that value of K and n always are
n
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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