ETC…..
From theses first ten terms of the sequence we can derive a formula for in terms of . The formula is:
By plotting the first ten terms of the sequence on a graph, we can study the relation between.
Graph 1
This illustrates the relation between n and L in the case of .
From this graph we can see that the value of L is slowly moving towards the value of aprox. 1,618, but the value of L will never attain this value. If we look at the relation between and we can establish that in the case of .
, when n approaches infinity,
lim() → 0
We can expand this and arrive at another infinite surd with a sequence of:
We can take this sequence and expand it:
ETC…
Graph 2
This graph illustrates G in the case of an.
From this graph we can clearly see that approaches the value of 2, but it never arrives at 2.
Now consider:
As we are dealing with an infinite surd, we can expand this out to get:
This can be expressed as:
From this we can determine:
Comparing this with our graph, we can conclude that our general term is correct!
