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Continued Fractions This 'infinite fraction' can be considered as a sequence of terms, tn: A general formula for tn+1 in terms of tn can now be determined. It can be seen that tn+1 is 1added to 1 divided by the previous term. i.e. Decimal equivalents of each term can be computed. Here are the values for the first ten terms (correct to 5d.p as we can then see how the numbers differ): t1=1 t2= 1.50000 t3= 1.66667 t4= 1.60000 t5= 1.62500 t6= 1.61538 t7= 1.61905 t8= 1.61765 t9= 1.61818 t10= 1.61798 From looking at the graph we can see that for the first few terms the values fluctuate, but eventually the values fluctuate less and become very close together. I.e. the values become closer together as the value of n increases. We can then conclude that as n increases tn˜tn+1. From this, we can now deduce a formula for tn+1 in terms of tn: We can also conclude that if tn˜tn+1, then tn-tn+1...

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