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Infinite Surds investigation with teacher's comments.

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Introduction

15th February, 2012

Portfolio Task 1

Type I – Infinite Surds

* Add headings

* Name figures and refer to them where necessary

In this portfolio, I will find the general statement that represents all the values of k for which the exact value of the infinite surd

k+k+k+k+k+…

 is an integer.  To do so, I will first investigate several different expressions of infinite surds in square root form and find the exact values of these surds.

A surd is the root of non-perfect powers.  It is an irrational number which exact value can only be expressed using the radical or root symbol is called a surd.  For instance, 2 is a surd because the square root of 2 is irrational.  An infinite surd is a never ending irrational number.  Its exact value could be left in square root form.

The term “surd” traces back to Al-Khwarizmi, an Arabic mathematician during the Islamic empire in or around 825AD who referred to rational and irrational numbers as “audible” and “inaudible” respectively.  Later, the Arabic “asamm” for irrational number was translated as “surdus” (meaning deaf or mute in Latin) by Gherardo of Cremona, a European mathematician in 1150.  Fibonacci (1202)

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Middle

 increases at a decreasing extent.  That is, each time it increases less than before.  The scatter plot seems to have a horizontal asymptote at y=1.618.  The value of an approaches approximately 1.618 (i.e. an≈1.618) but never reach it.

In other words, as n gets larger, the difference with the successive term (also shown in the figures in the “an+1 - an” column of the table) is gradually decreasing and almost reaches 0.

What does this suggest about the value of an- an+1 as n gets very large?

The above observation suggests that as n gets very large and at a certain large enough point, an will cease to increase and remain stable at a value.  The graph will then follow a straight horizontal line.

In other words, as n approaches infinity, the value of an- an+1 approaches 0.  To express in numerical notations, as n→∞, an=an+1.

Use your results to find the exact value for this infinite surd.

The above analysis so far only shows that an≈1.618.  To find the exact infinite value for this sequence, we need to rearrange the recursive formula an+1=1+an.

As mentioned, as n→∞, an=an+1.  The recursive formula becomes an=1+an.

(Alternatively, as n→∞

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Conclusion

When k = 6, …

When k = 12, …

When k = 20, …

When k = 30, …

Find the general statement that represents all the values of k for which the expression is an integer.

Explain how you arrived at your general statement.

The general statement that represents all the values of k would be:

For the expression 1+1+4k2 to be an integer, there are two conditions.

(1) 1 + 4k must be a perfect square.  This means that 1 + 4k is must be an integer larger or equal to 0, and its square root is a positive integer or 0.

(2) 1+1+4k must be an even number so that after it is further divided by 2, the final result would be an integer.

Test the validity of your general statement using other values of k.

To prove my general statement is valid, we consider 3 cases.

When both Conditions (1) and (2) are not met

k = 3

When only Condition (1) is met,

k = 3.75

(Note: There is no case where only Condition (2) is met because if 1 + 4k is not a perfect square, 1+1+4k would not be an integer.)

When both Conditions (1) and (2) are met

k = 20

Discuss the scope and / or limitations of your general statement.

Limitations: Meet above 2 conditions

* Further elaboration

1+4k has to be odd integer.

1 + 4k has to be square of odd integer.

k = n2 – n

Where n  Z+, n>/= 2

...read more.

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