- Level: International Baccalaureate
- Subject: Maths
- Word count: 1305
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.
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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.
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 where,, , and
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.
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Formula for an+1 In Terms of an
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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.
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A graph can be plotted using the data from the previous page.
Middle
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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.
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Relation of an and an+1
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 . The expression , used to describe the difference between an and an+1 when for the last example, is also true for in this case, proving that is also a convergent sequence.
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Solving the Sequence
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General Infinite Surd
Conclusion
We know that the calculations are true because . So this means that both the value of k and can be infinity. So the following conclusion can be made. , and
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:
- The relation between an and an+1 for all sequences of infinite surds can be represented by .
- The expression can be used to represent the exact value of all infinite surds. The restrictions are, , and that can only be a positive odd integer, 0 or .
- Infinite surds all have a convergence.
- The formula 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.
- In general, for the infinite surd to have a real solution, , and the value of the infinite surd when k is 0,, or a positive integer is .
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This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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