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.
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) →0
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:
→
x2=1+x →
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 ← 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
← 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.
