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# Infinite Surds. The aim of this folio is to explore the nature of infinite surds.

Extracts from this document...

Introduction

Mathematics Folio: Infinite Surds        Michael Zhao

4/04/2012

The aim of this folio is to explore the nature of infinite surds. Infinite surds take the form
.

Part one:

Consider the surd where:

Etc.

To begin, a recursive rule was determined from observing what changed from  to  and so on. The major change was that the previous surd was added to 1 and then square rooted to provide the next value. This observation provided the basis for the recursive rule below.

The decimal values were calculated for to  and the results were then graphed.

 n a(n) a(n) - a(n-1) 1 1.414214 -0.139560412 2 1.553774 -0.044279208 3 1.598053 -0.013794572 4 1.611848 -0.004273452 5 1.616121 -0.001321592 6 1.617443 -0.000408492 7 1.617851 -0.00012624 8 1.617978 -3.90113E-05 9 1.618017 -1.20552E-05 10 1.618029 -3.72528E-06

It can be observed that the slope of the function gradually decreases asymptotes towards a value of. The values of  suggest that as  becomes larger, the function asymptotes towards a certain value. Finding the exact value of the surd requires being able to solve the surd. Solving the surd was done in the following method.

The exact value of the infinite surd

Middle

A table of values is composed for values of n in addition to a graph to identify possible asymptotes.

 n a(n) a(n) - a(n-1) 1 1.847759065 -0.113811496 2 1.961570561 -0.028798893 3 1.990369453 -0.007221459 4 1.997590912 -0.001806725 5 1.999397637 -0.000451766 6 1.999849404 -0.000112947 7 1.999962351 -2.82371E-05 8 1.999990588 -7.05928E-06 9 1.999997647 -1.76482E-06 10 1.999999412 -4.41206E-07

It can be observed that the values seem to approach 2 as  becomes larger. The graph has a very similar shape to the graph in Part One but with a sharper turn and the values for  also have a similar pattern in that they gradually decrease and become very small as  becomes larger. However, the values are different in that they become more distinct. E.g. the big gaps are bigger, smaller gaps are smaller. To solve, a similar method to the method used in Part one will be used.

The exact value for the infinite surd is 2 where. These results correspond to the calculated results validating the observation that the infinite surd approached 2 with greater values of.

Part Three: The General Infinite Surd -

Consider the General Infinite Surd

Conclusion

The results are tabulated to see if there is a correlation between the value for  and the integer value of the surd.

 Integer Value 1 0 2 2 3 6 4 12 5 20 6 30 7 42 8 56 9 72 10 90

The table shows how there is a growth in the value of  which correlates directly to the integer values of the infinite surd. The correlation can be represented by the formula  or or with the General expression for Infinite surds.

When put into an excel spread sheet, and tested, the values for  all returned with the matching values for. Those values of  when substituted into the table of values for  showed values which all approached integers which were the original  values. Examples of these results are shown below.

 n a(n) 1 20.9879465773076 2 20.9997130117844 3 20.9999931669461 4 20.9999998373082 5 20.9999999961264 6 20.9999999999078 7 20.9999999999978

 n a(n) 1 13.9818002261237 2 13.9993499929862 3 13.9999767854445 4 13.9999991709087 5 13.9999999703896 6 13.9999999989425 7 13.9999999999622

 n a(n) 1 25.99028852413850 2 25.99981324017810 3 25.99999640846470 4 25.99999993093200 5 25.99999999867180 6 25.99999999997450 7 25.99999999999950

Infinite Surds and their traits are all closely interconnected with each other. The value of the infinite surd and the value for
in the infinite surd have a unique correlation which means that they can easily be calculated at any time as long as one variable is at hand. In addition, the general expression of an Infinite Surd can be used to calculate the value of any surd effortlessly.

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

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