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Examining Fibonacci sequence right end behaviour
What is the limiting value for the ratio of consecutive terms in the Fibonacci sequence? To the far right, does this sequence behave like a geometric sequence? x1 x r x2 X1.5 1,1,2,3,5,8, , For the left end behaviour, the first few numbers, the Fibonacci Sequence does not appear to be of a geometric type, but perhaps the right end behaviour, the list does appear to be geometric. We can predict that “r” is a positive number because each term in the sequence has the same sign. Lets analyze some data.
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r, ratio of consecutive terms
1,1,2,3,5,8,13,21,34,55,89,144,233,..... Complete the chart. n={2,3,4,5,6,7,8,....} x r, ratio of consecutive terms 1 2 3 4 5 6 7 8 9 10 11 Plot the data and determine r x Data appears to be oscillating as “r” converges towards ≈ 1.62 Thus the further you go to the right, the more the Fibonacci sequence behaves like a geometric sequence because “r” appears to be more constant.
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Recursive Formula for the Fibonacci Sequence.
All sequences are defined as To generate the next term of the Fibonacci sequence, add the two previous terms.
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Algebraic proof for “r”
Assuming Fibonacci is like a geometric sequence in its right end behaviour. Geometric sequences have Now, multiply both sides by Divide by “a”, as Simplify Interpret the exponents Quadratic formula gives Divide both sides by r ≈ or r≈-1.118, inadmissible r>0 as all terms are positive in the sequence.
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