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This portfolio will investigate the patterns and aspects of infinite surds. Technologies, graphs, and charts will be used in the process of the investigation, allowing the understanding of infinites surds to be more comprehensive.

Extracts from this document...

Introduction

Introduction

This portfolio will investigate the patterns and aspects of infinite surds. Technologies, graphs, and charts will be used in the process of the investigation, allowing the understanding of infinites surds to be more comprehensive.  In the beginning, two examples of infinites surds will be examined, and some similarities between the two may be found. Using the knowledge gained from the previous two examples, we will try to come up with some general statements and restrictions that are true for all infinite surds. First we will start by defining a surd.

image00.png

This is an example of an infinite surd, where identical surds are being added under the previous root repeatedly.

We can also turn this into a sequence whereimage01.png,image11.png, image19.png, and image29.png

To find the relationship between an and an+1, we can first find a formula for an+1 in terms of an, the substitution method along with the algebraic process can be used to find the formula.

1

Formula for an+1 In Terms of an

2

Decimal Values of the First Ten Terms of the Sequence

Below are the values of the first ten terms of the sequence accurate to the 9th place after the decimal point.

image43.png

image51.png

image58.png

image60.png

image02.png

image03.png

image04.png

image05.pngimage06.pngimage07.png

3

A graph can be plotted using the data from the previous page.

...read more.

Middle

n

7

Decimal Values of the First Ten Terms of the Sequence

Below are the values of the first ten terms of the sequence accurate to the 9th place after the decimal point.

image17.png

image18.png

image20.png

image21.png

image22.png

image23.png

image24.png

image25.pngimage26.pngimage27.png

8

Relation of an and an+1

image28.png

Once again, as the value of n increases, the value of an seems to increase less. This is similar to what has happened with the previous example. The chart below describes the difference between an and an+1 as the value of n increases.

Difference of an and an+1

an - an+1

Decimal value rounded to the exact billionth

a1-a2

-0.1138114958

a2-a3

-0.028798892

a3-a4

-0.007221459

a4-a5

-0.001806725

a5-a6

-0.000451767

a6-a7

-0.000112947

a7-a8

-0.000028237

a8-a9

-0.000007059

a9-a10

-0.000001765

Compare to the previous example, the difference between an and an+1 is less in this case. However, the same pattern occurred. As the value of n increases, the difference between an and an+1 decreases. This means that when the value of n is very large image09.png. The expression image30.png, used to describe the difference between an and an+1 when image12.png for the last example, is also true for in this case, proving that image31.pngis also a convergent sequence.

9

Solving the Sequence

10

General Infinite Surd

...read more.

Conclusion

image48.png, the value of the expression should also be infinity. This is shown by the calculations below. image49.png

We know that the calculations are true because image50.png. So this means that both the value of k and image46.png can be infinity. So the following conclusion can be made. image52.png, and image53.png

Conclusion

By first taking a look at two simple infinite surds, we are able to see some similarities in them. Then we are able to apply what we learnt into coming up with the general statements of infinite surds. Graphs and charts were used to aid and made the understanding of infinite surds easier. In conclusion the following general statements about surds can be made:

  1. The relation between an and an+1 for all sequences of infinite surds can be represented by image54.png.
  2. The expression image36.png can be used to represent the exact value of all infinite surds. The restrictions are, image55.png , and that image56.pngcan only be a positive odd integer, 0 or image34.png.
  3. Infinite surds all have a convergence.
  4. The formula image44.png can be used to calculate the values of “k” that make the infinite surd an integer. The product of any two consecutive integers will allow the infinite surd to be an integer.
  5. In general, for the infinite surd to have a real solution, image57.png, and the value of the infinite surd when k is 0,image34.png, or a positive integer is image59.png.

16

...read more.

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