• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Mathematics portfolio on Infinite Surd

Extracts from this document...

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.

image00.png

Consider this surd as a sequence of terms an where:

image01.pngimage01.png=image50.pngimage50.png=1.414213562

image61.pngimage61.png=image69.pngimage69.png=1.553773974

image02.pngimage02.png=image08.pngimage08.png=1.598053182

image17.pngimage17.png=image28.pngimage28.png=1.611847754

image34.pngimage34.png=image44.pngimage44.png=1.616121207

image45.pngimage45.png=image46.pngimage46.png=1.617442799

image47.pngimage47.png=image48.pngimage48.png=1.617851291

image49.pngimage49.png=image51.pngimage51.png=1.617977531

image52.pngimage52.png=image53.pngimage53.png=1.618016542

image54.pngimage54.png=image55.pngimage55.png=1.618028597

According to the result, you can aware that a2 = image56.pngimage56.png

Then analyzing the formula, an+1 = image57.pngimage57.png

image58.png

On the graph, it represents that at the point of the beginning it raises rapidly as acceleration. However, after that, an-an+1

...read more.

Middle

So,

1.618033989

Consider another infinite surd

image66.pngimage66.png where the first term is image67.pngimage67.png

Repeat the entire process above to find the exact value for this surd.

image68.pngimage68.png=image67.pngimage67.png=1.847759065

image06.pngimage06.png=image70.pngimage70.png=1.961570561

image71.pngimage71.png=image72.pngimage72.png=1.990369453

image73.pngimage73.png=image74.pngimage74.png=1.997590912

image75.pngimage75.png=image76.pngimage76.png=.1999397637

image77.pngimage77.png=image78.png=1.999849404

image79.pngimage79.png=image80.pngimage80.png=1.999962351

image81.pngimage81.png=image82.pngimage82.png=1.999990588

image83.pngimage83.png=image03.pngimage03.png=1.999997647

image04.pngimage04.png=image05.pngimage05.png=1.999999412

According to the result, you can aware that image06.pngimage06.png =image07.pngimage07.png

Then analyzing the formula, image09.pngimage09.png=image10.pngimage10.png

image11.png

This graph shows us that there is a huge change between 1 and 2. And it is constant from 4.  . bn and image12.pngimage12.png-image09.pngimage09.png

...read more.

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 image13.pngimage13.png

With using quadratic equation,

image37.png

8x9=72----y = image38.pngimage38.png

=46

88x89=7832------image39.pngimage39.png

=89

222x223=49506---------image40.pngimage40.png

=223

And so on..

It is able to notice that

When k is 72, y = image41.pngimage41.png =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

image42.png

nimage43.pngimage43.png

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. IB Mathematics Portfolio - Modeling the amount of a drug in the bloodstream

    Therefore, the amount of drug left in the bloodstream before the next dose would get a little higher because more drugs is added before all the drug was gone in the bloodstream.

  2. Math Portfolio: trigonometry investigation (circle trig)

    and tan equaling Tan ?= y/x equaling sin ?( r ) divided by Cos ?( r ) then as what I said in the above, the r gets simplified just leaving. y= sin ?( r ) Tan ?= y/x equaling sin ?( r ) divided by Cos ?( r )

  1. Math Portfolio Type II

    + 2.56]un = (-2.6 x 10-5)un2 + 2.56un To find out the changes in the pattern, let us see the following table which shows the first 20 values of the population obtained according to the logistic function un+1= (-2.6 x 10-5)un2 + 2.56un just calculated: - n+1 un+1 n+1 un+1

  2. Math HL portfolio

    - (- ) = | 0.2386 - (2.2386) | = | -2 | = 2 D= = = 0.5 My conjecture is not working if the lines are changed !!! Now I am trying to modify my conjecture by analysing patterns which are occurring when the Parabola and/or the lines are changed.

  1. Math Portfolio type 1 infinite surd

    19 1.618033989 20 1.618033989 21 1.618033989 22 1.618033989 23 1.618033989 24 1.618033989 25 1.618033989 Notice: 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)

  2. Infinite surds portfolio - As you can see in the first 10 terms of ...

    Just like before, you can represent this whole infinite surd with: x = (x)2 = 2 Square both sides to create an equation to work with. x2 = The infinite surd continues. x2 = k + x Substitute with x since x =.

  1. Infinite Surds investigation with teacher's comments.

    * Need elaboration an+1=k+an … x=k+x. So, x2=k+x x2-x-k=0 Using the quadratic formula, x =-b±b2-4ac2a=-(-1)±(-1)2-41(-k)2(1)=1±1+4k2 1+1+4k2 for every value of k ∈ Q+ ________________ An integer is an entire number, and can be written without a fractional or decimal component.

  2. Infinite Summation- The Aim of this task is to investigate the sum of infinite ...

    The complete data collection for the equation, suggests that when n is ?, when x=1 and a =3, the sum will equal to 3. This may suggest that the domain for the function is 1?Sn?3. Now a general sequence where x=1, will be considered.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work