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Introduction

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 term.

Graph 1.

Middle

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 tnand tn+1gets so small that we can ignore the difference. And as the difference between tnand tn+1gets 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.

 n 1 2 3 4 5 6 7 8 9 10 tn          tn 3.00 2.33 2.43 2.41 2.41 2.41 2.41 2.41 2.41 2.41

Just as the

Conclusion

 n 1 2 3 4 5 tn     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 tnand tn+1decrease until it is so small that we can ignore it. And as the difference between tnand tn+1gets 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.          This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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