# Portfolio: Continued Fractions

Continued fractions

A continued fraction is a fraction whose numerator is an integer and whose denominator is an integer added to a fraction whose numerator is an integer and whose denominator is an integer added to a fraction, and so on.

The continued fraction above is the first continued fraction to consider. But how do the ten first terms look like, and how does the relationship between them look like?

Table 1. The ten first terms of the continued fraction above.

The ten first terms of the sequence, which can be seen in Table 1, led me to the generalized formula for the sequence.

However this sequence is not just a regular sequence, it is the Fibonacci sequence. And the Fibonacci sequence is a recursive sequence, which is a sequence in which a general term is defined as a fraction of one or more of the previous term.

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 ...