Fibonacci and Lucas Sequences

By Karl Mealor

The Fibonacci sequence is the sequence 1, 1, 2, 3, 5, 8, 13, 21,... where each subsequent term is generated by adding the two previous terms together. Lucas sequences are similar to the Fibonacci sequences, but the first two terms of Lucas sequences would be some values other than 1 and 1. The following discussion will examine the ratios of adjacent terms of Fibonacci and Lucas sequences.

To begin, click on the link below to examine the Fibonacci sequence as well as the Lucas sequence generated by the first two terms 3 and 7.

Column A lists the first 30 terms of the Fibonacci sequence.

Column B lists the ratios of adjacent terms of the Fibonacci sequence. (Each term is divided by the previous term.) Notice that the ratio reaches 1.61803399 which is a decimal approximation for the golden ratio. Click here for an explanation.

Column C lists the ratios of every other term. This ratio reaches 2.61803399 which is 1 more than the decimal approximation for the golden ratio as well as the square of the golden ratio. Why?

Column D lists the ratios of every third term. This ratio settles at 4.23606798 which is the decimal approximation for the cube of the golden ratio. Why?

Columns F, G, and H list the first 30 members of the given Lucas sequence, the ratios of adjacent terms and the ratios of every other term, respectively. Notice that the ratio of adjacent terms also approaches the golden ratio as expected. (Click here for an explanation.) The ratio of every other term also approaches the square of the golden ratio. Why?