The Fibonacci numbers are a well-known sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... ., where a subsequent number in the sequence is found by adding the two previous numbers together. The ratio of two consecutive numbers in the sequence approaches the Golden Ratio. For example 89/55=1.61818, 144/89=1.61798 and 233/144=1.61806. The ratios alternate around the Golden Ratio and get closer and closer to Φ as the numbers increase.
The Lucas numbers are a sequence built with the same rule as the Fibonacci numbers except instead of starting with 1 and 1, it starts with 1 and 3. The first few terms are: 1, 3, 4, 7, 11, 18, 29, 48, 77, 125, 202, 327... . Interestingly, the ratio between consecutive numbers as the sequence grows is still approaching the Golden Ratio. 77/48=1.60417, 125/77=1.62338, 202/125=1.61, 327/202=1.61881. Even more astonishing is the fact that you can choose any two numbers at random, with any number of digits, add them together and build a sequence off of them using the same rule as the Fibonacci and Lucas numbers. In doing so you will obtain a close approximation to the Golden Ratio by using about the 19th and 20th terms in the sequence! (Livio, 2003) (To see this view the applet here )
The Golden Ratio has its own unique sequence. Amazingly it is simultaneously an additive sequence (a sequence obtained by beginning with 1 or 2 terms and then using a rule of addition or subtraction repeatedly to get the subsequent terms of the sequence) and a geometric sequence (sequence which has a constant ratio between each two sequential terms in the series). The Golden Sequence is the only one there is that is both. One way to write the sequence is: 1, Φ, 1 + Φ, 1 + 2Φ, 2 + 3Φ, 3 + 5Φ, 5 + 8Φ, 8 + 13Φ,... (each subsequent term is found by adding the two previous ones together) and since 1 + Φ = Φ2, another way to write the sequence is: 1, Φ, Φ2, Φ3, Φ4, Φ5, Φ6,... (each subsequent term is found by multiplying the previous term by Φ) (Huntley, 1970).
There is an interesting problem made famous by Charles Lutwidge Dodgson (1832-1898) whose penname was Lewis Carroll (Posamentier, 2007). It illustrates how the Fibonacci numbers and the Golden Sequence differ in a "geometric fallacy". Use the applet here to investigate this. The fallacy lies in the misalignment of the center diagonals. Here are an activity plan and some questions for group discussion to go along with the applet "Geometric Fallacy".