Richard Francisco

The Number of Assignments for this Course is Growing Like Families of Rabbits!

The table at the bottom of this page contains values of the Fibonacci (1st column) and Lucas (3rd column) sequences. The Fibonacci sequence is a sequence where the first two values are equal to one, and each successive term is defined recursively, namely the sum of the two previous terms. The Lucas sequence is similar, though the first term is one and the second term is three, but defined equivalently with the Fibonacci sequence thereafter.

The entries in the second and fourth columns are the ratios of the two preceding terms in the respective sequence. These entries appear to approach the same number, which would be the limit of the ratio of the terms. I will prove that there is such a limit, and give the value of this number.

To consider the limit of the Fibonacci sequence, let

by the properties of limits,

It will be helpful to explicitly state the construction of the Fibonacci sequence to manipulate the above expressions:

Using this equation to substitute, we get

and so we get the equivalent equations

and finally solving for L using the quadratic formula yields:

which is the limit of the ratio of the terms, and is approximately 1.618034

As a fun fact, the explicit formula of the Fibonacci sequence is:

This table confirms the above calculations.

 1 1 1 1 1 1 3 3 2 2 4 1.33333 3 1.5 7 1.75 5 1.66667 11 1.57143 8 1.6 18 1.63636 13 1.625 29 1.61111 21 1.61538 47 1.62069 34 1.61905 76 1.61702 55 1.61765 123 1.61842 89 1.61818 199 1.61789 144 1.61798 322 1.61809 233 1.61806 521 1.61801 377 1.61803 843 1.61804 610 1.61804 1364 1.61803 987 1.61803 2207 1.61804 1597 1.61803 3571 1.61803 2584 1.61803 5778 1.61803 4181 1.61803 9349 1.61803 6765 1.61803 15127 1.61803 10946 1.61803 24476 1.61803 17711 1.61803 39603 1.61803 28657 1.61803 64079 1.61803 46368 1.61803 103682 1.61803 75025 1.61803 167761 1.61803 121393 1.61803 271443 1.61803 196418 1.61803 439204 1.61803 317811 1.61803 710647 1.61803 514229 1.61803 1149851 1.61803 832040 1.61803 1860498 1.61803 1346269 1.61803 3010349 1.61803 2178309 1.61803 4870847 1.61803 3524578 1.61803 7881196 1.61803 5702887 1.61803 12752043 1.61803 9227465 1.61803 20633239 1.61803 14930352 1.61803 33385282 1.61803 24157817 1.61803 54018521 1.61803 39088169 1.61803 87403803 1.61803 63245986 1.61803 141422324 1.61803 102334155 1.61803 228826127 1.61803