1. 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.

2. 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.

3. 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.

4. 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.