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

Infinite surds Maths Portifolio

Extracts from this document...

Introduction

Mathematics SL

Portifolie

Infinite Surds

An infinite surd is a never-ending positive irrational number. It is a number that can only be expressed exactly using the root sign image00.png.

image01.png

This sequence above is known as an infinite surd and can be expressed in the terms ofan:

a1 = image11.png= 1.414213

a2 = image06.png= 1.553773

a3= image30.png= 1.598053

a4 = image40.png= 1.611847

a5 = image49.png= 1.616121

a6= image56.png= 1.617442

a7= image57.png= 1.617851

a8= image58.png= 1.617977

a9= image02.png= 1.618016

image03.png

etc.

This is the first ten terms and the formula for these sequences is:

image04.png

because if we use the term a2 as an example, this could be proven as:

image04.png

a1+1 = image05.png

a2 =image06.png

a2 = 1.553773

By plotting a graph of the ten first term of this sequence the relationship between n and an could be shown:

image07.png

As can be seen from this graph is that the values increase, but then flattens out. The values of an moves towards the value of 1.618 approximately, but will actually never reach it. This can be understood by:

an - an+1

as n gets very large.

lim(

...read more.

Middle

 = 1.990369453

b4 = image15.png = 1.997590912

b5 = image16.png= 1.999397637

b6 = image17.png= 1.999849404

b7 = image18.png= 1.999962351

b8 = image19.png= 1.999990588

b9 = image20.png= 1.999997647

image21.png

By replacing an by bn the formula for this sequence is:

bn+1 = image22.png

because if we use b2 as an example, the answer will be:

b1+1 = image23.png

b2 = image13.png

b2 = 1.961570561

By plotting a graph, the relationship between n and bn can be shown:

image24.pngimage25.png

As can be seen from this graph is that the value of bn increases as n gets larger. After a while the curve flattens out, but continue to rise. The values of bn approaches 2, but will actually never reach it. This can be understood by:

bn – bn+1

As n becomes very large, the value approaches 0 because:

lim(bn – bn+1) → 0

When n becomes very large and reaches infinity, the value approaches 0. This is because the difference between the two values is so small that the difference becomes insignificant.

The exact value of this infinite surd is:

bn+1 can also be written as bn so:

bn = image26.png

...read more.

Conclusion

k = 2.1

an = image45.png

an = image46.png

an = 2.03297

Ex. 6

k = 2.5

an = image47.png

an = image48.png

an = 2.15831

As can be seen from example number 1 is that if k is a negative number we get square root of a negative number and therefore there will be no solution. Because of this, k cannot be a negative number.

From example 2 and 4 this can be found: There are few numbers that gives the answer of an integer in this sequence expressed with k. Some of the values are the numbers 0, 2, 6, 12 and 20 etc. From this we can see an increasing trend were:

0 = 2image50.png

2 = 2image51.png

6 = 2image52.png

12 = 2image53.png

20 = 2image54.png

We can see that from 0 to 2 the number we multiply by the constant, 2, increases by 1, and from 2 to 6 it increases by 2. From 6 to 12 it increases by 3, and from 12 to 20 it increases by 4.

If we use other numbers from 0 to 20 than those above, we get decimal numbers. This can be seen from example 3, 5 and 6. As k increases, the value of an increases as well.

The scope is 0< n >image55.png, because n cannot be a negative number.

...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. Infinite Surds Investigation. This graph illustrates the same relationship as was demonstrated in the ...

    those what numbers, when divided by two, would make the expression an integer. So, to find values that would make k a perfect square, I would set: , , , and so on... Example Solution for k: This table of values shows several the solutions for k using perfect square:

  2. Math IA, Infinity surd

    Obviously, "K" value is positive. We now have worked out the exact value(In other words, the Limit), therefore we know that the number will be keep increasing so close to the number, but it will never reach it. Consider another infinity surd The first term is given, .

  1. Stellar Numbers. In this task geometric shapes which lead to special numbers ...

    as in the quadratic equation: ax2 + bx + c The value of 'a' is half the constant difference. In this example a= Now that I know that the first part of the formula is n2 I can proceed to find the values of 'b' and 'c'.

  2. Infinite Summation - In this portfolio, I will determine the general sequence tn with ...

    + = 6.067246 S4 = S3 + t4 = 6.067246 + = 6.664666 S5 = S4 + t5 = 6.664666 + = 6.897171 S6 = S5 + t6 = 6.897171 + = 6.972577 S7 = S6 + t7 = 6.972577 + = 6.993539 S8 = S7 + t8 = 6.993539

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

    you can see that the greater n increases, the closer an gets to the value of about 1.61803. But an never touches that value; that causes the curve to flatten out. It also shows that the rise of the slope is continually decreasing as n increase.

  2. MAth portfolio-Infinite surds

    In the general infinite surd: a1 = a2 = also, a2 = a3 = Squaring both sides, (a3)2 = ()2 (a3)2 = a2 + k a2 = (a3)2 - k Therefore, an-1 = (an)2 - k so, k = (an)2 - an-1 Q5)

  1. Infinite Surds investigation with teacher's comments.

    are plotted in the graph below showing the relationship between n and an. * Specify technology? (Graph) By plotting the relation between n and an, we can observe that there is a positive relationship between n and an. That is, as n increases, an also increases.

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

    Now a general sequence where x=1, will be considered. Taking all data collected into account a relation between a and ? has shown. So as the two sets of results show a=2 or a=3, the total of sum won?t exceed 2 or 3, when x=1. To prof this suggestion right, two more examples will be made, again when x=1

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