Portfolio: Continued Fractions

The first 200 words of this essay...

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.

n

1

2

3

4

5

6

7

8

9

10

tn

tn

2.00

1.50

1.67

1.60

1.63

1.62

1.62

1.62

1.62

1.62

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