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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Introduction
Math Portfolio
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
a5
1.616121207
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)
Middle
gets closer to a fixed value.
...read more.
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) →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
Conclusion
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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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