infinte surds

8/7/2019 infinte surds

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Vivian Li 12C26th March, 2008

Introduction:

In this portfolio, I am going to investigate the infinite surds. For

example the sequence of the following surd:

a1=

a2=

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a3= etc

First I will find a general formula for the surd above, then I willinvestigate the relationship between n and an. After had try and

explain one set of surd, I will then set another set of surd and

investigate it. After that, I will consider the general infinite surd

and find an expression for the exact value of the general infinite

surd in terms of k. Then I will test the validity of the general

statement using other values of k, and I will discuss the scope

and limitations of the general statement.

The decimal values of the first ten terms of the sequences:

a1= = 1.414214

a2= = 1.553774

a3= = 1.598053

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a4= = 1.611848

a5= = 1.616121

a6= = 1.617443

a7= = 1.617851

a8= = 1.617978

a9= = 1.618017

a10= = 1.618029

n an+1 n an+11 1.553774 11 1.6180323232 1.053 12 1.6180334743 1.611848 13 1.618033830

4 1.616121 14 1.6180339405 1.617443 15 1.6180339746 1.617851 16 1.618033984

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7 1.617978 17 1.6180339878 1.618017 18 1.6180339889 1.618029 19 1.61803398910 1.618032 20 1.618033989

The general formula for an+1 in terms of an will be:

an+1= (By inspection)

After calculating the value of an+1, a graph Is then be plotted.

Graph of an+1=

Through Fig 1.1, we can see the values of an is increasing rapidly

for the first few terms, then the values of an approaches to a certain

value as we can see in the graph. As n gets larger, the limit of the

surd is closes to 1.62, also we can see from the graph, the values of

an- an+1 is very close to zero. This suggest that:

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Fig 1.1

Fig 1.1


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The exact value of this infinite surd will be:

Let an+1= an= xX=

X2=

(1.62) OR (Rejected)

The reason of being rejected is because the sum of

positive numbers cant be negative number.

After had try and explain one set of surd, I will then set another

set of surd to investigate. This time I will try

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where the first term is .

a1=

a2=

a3=

a10=

Graph of an+1=

As we can see in Fig 1.2, the graph is look similar to Fig 1.1.

When the values of an is getting larger, the limit of the surd is closes

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Fig 1.1

Fig 1.2


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to 2, also we can see that the difference between an and an+1 is very

close to zero.

General formula:

an+1=

Again let an+1= an= x

As I have proved above, when n gets larger, the limit

of the surd is closes to 2. This suggest that an-an+1 is

very close to zero.

X=

X2=

X= 2 OR X= -1 (Rejected)

The reason of X= -1 being rejected is because the sum

of two positive numbers cant be negative number.

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After had tried and explain two sets of surds, I will then consider

the general infinite surd in terms of k, to find an expression for the

exact value of this general infinite surd.

a1=

a2=

a3=

a10=

General Formula:

Let an+1= an= x

OR (Reject)

Which , when the result must be an integer.

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If the solution must be an integer then , where M

is any positive integer 1.

Product of any 2 positive consecutive numbers.

The general statement that represents all the values of K for which

the expression is an integer is

Now, I will use different value of M to test whether my statement is

correct or not.

Example 1:

If M=2

K= 2

Substitute K into the formula of the exact value of the general

infinite surd

X= 2, which is an integer.

Example 2:

If M= 3

K= 6


infinite surd

X= 3, which is an integer

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Example 3:

If M= 5

K= 20


infinite surd

X= 5, which is an integer

Example 4:

If M= 100

K= 9900


infinite surd

X=100, which is an integer

The four examples above has proved that if K is the product of any 2

positive consecutive numbers, then the exact value of the infinite

surd must be an integer.

But if K is not the product of 2 positive consecutive numbers, then

the exact value of the infinite surd will not be an integer. Here are

the examples:

Example 1:

If K is rational number, example like , then substitute K into the

formula of the exact value of the general infinite surd

X= 1.2071(4 decimal place), which is not an integer

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Example 2:

If K is a square root number, example like , then substitute K into

the formula of the exact value of the general infinite surd.

X=1.79(two decimal place), which is not an integer

Example 3:

In K is irrational number, example like , then substitute K into the

formula of the exact value of the general infinite surd.

X= 2.342(3 decimal place), which is not an integer.

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Conclusion:

In this portfolio, we are investigating the infinite surds

. We substitute K=1 and K=2 to find the general

formula and the equation of calculating the exact value. And we find

that the general equation of this is and

the exact value formula is . When K=0, the result of the

infinite surd is equal to zero as expected. For any K bigger then 0,

there is always a limit for the infinite surd. If , where M

is any positive integers, the infinite surds will be an integer equal to

M. Otherwise, the result will be an irrational number.

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