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Fibonacci Ratios

The Fibonacci Sequence starts with the numbers 1,1... each successive number is found by adding the two numbers the precede it. What happens to the ratio of one Fibonacci number to the number just before it as the numbers increase in the sequence? (Why?)
How about if you start with numbers other than 1 and 1? Pick ANY two starting numbers enter them in the a_1 and a_2 boxes below: