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.