- Level: International Baccalaureate
- Subject: Maths
- Essay length: 663 words
Infinite Surds. As we can see there are ten terms of this sequence where is the general term of the sequence when, is the first term of the sequence.
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Infinite Surds
Introduction
a surd is an irrational number that can not be written as a fraction of two integers but can only be expressed using the root sign.
Bellow an example of an infinite surd:
This surd can be turned into a set of particular numbers sequence:
1.414213562
1.553773974
1.598053182
1.611847754
a51.616121207
1.617442799
1.617851291
1.617977531
1.618016542
As we can see there are ten terms of this sequence where is the general term of the sequence when, is the first term of the sequence...Etc.
A formula has been defined for in terms of:
(1)
A graph has been plotted to show the relation between and. And it can be oticed that as long as gets larger, gets closer to a fixed value.
To investigate more about this fixed value we take this equation into consideration as n gets bigger.
n
an-an+1
1
- 0.13956
2
-0.04428
3
-0.01380
4
-0.00427
5
-0.00132
6
-0.00040
7
-0.00013
8
-0.00004
9
-0.00001
We can figure out from the table above that when n gets larger, the term (an+an+1) gets closer to zero but it never reaches it
So we can come to the conclusion:
When n approaches infinity, lim (an-an+1) ï¿½0
An expression can be obtained in the case of the relation between n and an to get the exact value of the infinite surd:
= x
If we apply formula (1) to this:
ï¿½
x2=1+x ï¿½
The equation can be solved using the solution of a quadric equation:
Whereas a=1, b=-1, c=-1
Two solutions for x were obtained:
x=1.618033989 and x= -0.61803387
The negative value is ignored so x=1.618033989 which is the exact value for this infinite surd.
Another condition of infinite surd can be taken into consideration to acknowledge the point more:
Where the first term of the sequence is
The first ten terms of the sequence are:
1.847759065
1.961570561
1.990369453
1.997590912
1.999397637
1.999849404
1.999962351
1.999990588
h10= 1.999999421
And the formula of the sequence is obtained according to bn+1 which is relative to the term bn:
The graph bellow shows the relation
It can be observed that when n gets bigger, bn attempt to reach the value 2 which is the exact value for this infinite surd.
To prove that 2 is the exact value an expression is used:
Where x= exact value
x=
(x-2)(x+2)=0
x=2
x=-2
Since only the positive value is concerned then x=2 which is the exact value for the infinite surd.
Now we think about a general infinite surd to prove our previous work.
We consider the general infinite surd as:
Now let x =
Squaring both sides
x2=k+x
x2 - x - k=0 ï¿½ the expression for the exact value of the general infinite surd
A general statement could be found to make the expression an integer, and its be solving the equation above using the solution of a quadric equation:
The negative solution is ignored so:
To find some values of k to make the expression an integer:
We can see that 4k is an even number and 4k+1 is odd, so is an odd number if 4k+1 is a perfect square hence 1+ is an even number and possible to be divided by 2. As a result if 4k+1 is a perfect square we can obtain an integral number in the result.
For example let k=2:
k=3
ï¿½ not integer because13 is not a perfect square
Thus we come to the conclusion that only limited values of k can be used to make the result an integer and those values are any value of k can make 4k+1 a perfect square such as k = 2,6,12...etc.