• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Infinite Surds investigation with teacher's comments.

Extracts from this document...

Introduction

15th February, 2012

Type I – Infinite Surds

* 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)

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→∞

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

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## The Fibonacci numbers and the golden ratio

By changing the start numbers I proved that it doesn't make a difference which numbers we use. To prove my conjecture I've changed the equation and solved it. Afterwards I found the discriminant (5), which leads me to find the roots.

2. ## This portfolio will investigate the patterns and aspects of infinite surds. Technologies, graphs, and ...

In this case, the sequence would be expressed as ,, , etc. Now we will try to find a formula for an+1 to show the relationship between an and an+1. Formula for an+1 In Terms of an 7 Decimal Values of the First Ten Terms of the Sequence Below are

1. ## This essay will examine theoretical and experimental probability in relation to the Korean card ...

Gget-8, Gget-7, Gget-6, Gget-5, Gget-4, Gget-3, Gget-2 and Gget-1, the experimental probabilities were closer to the theoretical values. These hands had less than 20 percentage difference between the theoretical and experimental probability. Also, there were hands that had a percentage difference which were greater than 20 percent.

2. ## Math Portfolio: trigonometry investigation (circle trig)

tan turn out to be positive while the values of sin and cos turn out to be negative. When we put a random angle from quadrant 4, the range of -90<?<0, -7� in trial to verify the conjecture, the value of cos turns out to be positive and the values of sin and tan turn out to be negative.

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

The value of an infinite surd is not always an integer. Find some values of that make the expression an integer. Find the general statement that represents all values for is for which the expression is an integer. The first value of which is an integer is 2 which is

2. ## Maths Project. Statistical Analysis of GCSE results at my secondary school summer 2010 ...

graph but this graph is showing the grades against the Alphabetical positions base on Mathematic results only and as we can see on the graph there isn't any trend (line of best fit) on the graph, so placing one there will be useless.

1. ## Infinite Summation- The Aim of this task is to investigate the sum of infinite ...

and x using the gained results. (Diagram Microsoft Excel) This diagram shows the same results as when T9(2,x) and T9(3,x), so that when x increases, the value of T7(10,x) will increase too. As x increases the curve slopes up regularly. For the first example, a=e when Tn is T7, T7(e,1)

2. ## Parallels and Parallelograms Maths Investigation.

á´ A8, A4 á´ A9, A5 á´ A6, A5 á´ A7, A5 á´ A8, A5 á´ A9, A6 á´ A7, A6 á´ A8, A6 á´ A9, A7 á´ A8, A7 á´ A9, A8 á´ A9. Adding a tenth transversals gives us a total of forty-five parallelograms. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 