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# 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.

Extracts from this document...

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

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)

Middle

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) →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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