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Maths Portfolio Infinite Surds

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Introduction

Maths Portfolio

SL Type 1

Infinite Surds

In this mathematics portfolio we are instructed to investigate different expression of infinite surds in square root form and then find the exact value and statement for these surds.

INFINITE SURDS

The following expression is an example of an infinite surd.

image05.png

The first ten terms of the surd can be expressed in the sequence:

image06.png

From the tem terms of the sequence we can observe that the formula for the sequence is displayed as:


image16.png

The results had to be plotted in a graph as shown below:

image27.png

The graph above shows us the relationship between n andimage10.png. We can observe that as n increases, image10.png also increases but each time less than before, suggesting that at a large certain point of n it stops increasing and just follows in a straight line.

...read more.

Middle

From the tem terms of the sequence we can observe that the formula for the sequence is displayed as:


image08.pngimage09.png

The results had to be plotted in a graph as shown below:

The graph above also shows us the relationship between n andimage10.png. We can observe that as n increases, image10.png also increases but each time less than before, suggesting that at a large certain point of n it stops increasing and just follows in a straight line.

To find the exact infinite value for this sequence we would use the equation and rearrange in the correct way to find the value.

Finding the exact value for the surd

image00.pngimage11.png

image03.pngimage04.png

image12.png

image13.png

The exact value for the surd is 2 as the second answer does not fit in the problem

The general infinite surd in terms of k

image14.png

...read more.

Conclusion

image30.pngimage29.png

The results show us that the limitations found in the general statement are those that didn’t give an integer as a final answer. Therefore k has to be an integer, however not a decimal or a fraction as results wouldn’t be a whole number. It also isn’t possible to obtain an integer if k is a negative number because there is no square root for a negative number. For a pleasing result, 1+4k , needs to give a number that possesses a perfect square root, an integer.

I arrived at this general statement by subtracting the n term from the k, and like that finding out that the product of this subtraction is the n term to the power of 2. Like that I found the equationimage31.png. By rearranging the formula to make k the subject I came to the conclusion that image32.png, the general statement.

...read more.

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

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