Graph 1. The value of the terms versus the term numbers.
As n increases does the difference between the value of a term and the value of the term before decrease. This towards a specific value, and since we know that the sequence is the Fibonacci sequence we also know that the specific value the terms is moving towards is the golden ratio.
When looking for the exact value of, for example, the 200th term problems arise. Mostly because of the fact that it is hard and takes a lot of time to count to the 200th term by hand, but also because the difference between the 200th term and the 199th term is so small that it is hard, even with a computer, to find a value of the 200th term which has enough significant figures to differ from the 199th term.
However as n goes further and further on towards infinity, tn goes further and further towards the golden ratio. And as n gets very large the difference between tn and tn+1 gets so small that we can ignore the difference. And as the difference between tn and tn+1 gets so small that we can ignore it, we can call both of the terms for t and therefore rewrite the generalized formula for the Fibonacci sequence.
Now we can establish an exact value for the continued fraction with help from the quadratic formula and the quadratic equation, and the value we will establish is the golden ratio.
The second continued fraction to consider is the continued fraction below.
Table 2. The ten first terms of the continued fraction above.
Just as the other continued fraction we have considered, the Fibonacci sequence, does this continued fraction have a generalized formula; however it does not look exactly like the generalized formula of the Fibonacci sequence, it does instead looks like the one below.
Graph 2. The value of the terms versus the term numbers.
In Graph 2 we can see that as n goes further and further on towards infinity, tn goes further and further towards a specific value; just as the Fibonacci sequence went towards the golden ratio.
And as n gets very large the difference between tn and tn+1 gets so small that we can ignore the difference. Once again just as in the case of the Fibonacci sequence. And as the difference between tn and tn+1 gets so small that we can ignore it, we can call both of the terms for t and therefore rewrite the generalized formula for this continued fraction.
Now we once again can use the quadratic formula and the quadratic equation to establish the exact value of the continued fraction, the value the continued fraction is moving towards.
Now the general continued fraction.
Table 3. The five first terms of the continued fraction above.
Just as the two other continued fraction we have considered, does the general continued fraction have a generalized formula, and it looks like the one below.
Now when I have established the generalized formula for the general continued fraction, I can, once again with help from the quadratic formula and the quadratic equation, establish a general formula for the exact value of the general continued fraction. Because this continued fraction does also go towards a specific value, the exact value of the continued fraction.
And as n goes further and further on towards infinity and gets very large does the difference between tn and tn+1 decrease until it is so small that we can ignore it. And as the difference between tn and tn+1 gets so small that we can ignore it, we can once again call both of the terms for t and therefore rewrite the generalized formula for the general continued fraction, and establish the exact value of the general continued fraction.
