Download - infinte surds
-
8/7/2019 infinte surds
1/12
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=
1
-
8/7/2019 infinte surds
2/12
Vivian Li 12C26th March, 2008
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
2
-
8/7/2019 infinte surds
3/12
Vivian Li 12C26th March, 2008
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
3
-
8/7/2019 infinte surds
4/12
Vivian Li 12C26th March, 2008
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:
4
Fig 1.1
Fig 1.1
-
8/7/2019 infinte surds
5/12
Vivian Li 12C26th March, 2008
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
5
-
8/7/2019 infinte surds
6/12
Vivian Li 12C26th March, 2008
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
6
Fig 1.1
Fig 1.2
-
8/7/2019 infinte surds
7/12
Vivian Li 12C26th March, 2008
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.
7
-
8/7/2019 infinte surds
8/12
Vivian Li 12C26th March, 2008
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.
8
-
8/7/2019 infinte surds
9/12
Vivian Li 12C26th March, 2008
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
Substitute K into the formula of the exact value of the general
infinite surd
X= 3, which is an integer
9
-
8/7/2019 infinte surds
10/12
Vivian Li 12C26th March, 2008
Example 3:
If M= 5
K= 20
Substitute K into the formula of the exact value of the general
infinite surd
X= 5, which is an integer
Example 4:
If M= 100
K= 9900
Substitute K into the formula of the exact value of the general
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
10
-
8/7/2019 infinte surds
11/12
Vivian Li 12C26th March, 2008
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.
11
-
8/7/2019 infinte surds
12/12
Vivian Li 12C26th March, 2008
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.
12