Fibonacci Sequence

"Phi, or µ, is a representation of a number which, when multiplied by itself, is the same as itself plus 1 [ µ2- µ-1=0]. µ = 1.618033989. µ2 = 2.618033989. µ+1 = 2.618033989." (Conklin). The exact representation of Phi is (1+áÌ5)/2. Going back to the Fibonacci sequence, which begins with 1, 1, 2, 3, 5, we are able to make two observations. The first is that if one finds the ratios of the successive numbers, the results converge on 1.618, thus: 1/1=1, 2/1=2, 3/2=1.5, 5/3=1.67, 8/5=1.6 The second is when one follows next equations, the Fibonacci numbers become apparent: µ1=1.618, µ2=( µ*1)+1, µ3=( µ*2)+1, µ4=( µ*3)+2, µ5=( µ*5)+3, and so on. The most direct relationship between Phi and Fibonacci numbers is that Phi is the limit of b/a. In other words, if one was to calculate Fibonacci numbers to the infinity, take the last number (b) and divide it by the previous number (a), that would be the exact representation of Phi. Due to the fact that one cannot reach the infinity, the limit of b/a tells us that Phi is an asymptote of the equation which calculates Fibonacci numbers. Since Phi is defined as a number which, when squared, is the same as that number plus one, an important questions comes up, is Phi unique? In other words, is there another number that follows the same pattern? As far as the modern mathematics, no one was able to find a similar number, therefore, the properties of Phi are considered unique. What this demonstrates is that "Phi is not this number that sits by itself, but that it is part of a larger pattern of number sequences" (Conklin), the Fibonacci Numbers. Another simple, yet important note that needs to be made about Phi, is that as it is accepted in mathematics, Phi and phi are actually two different numbers. Though closely related, phi is actually Phi-1. Though this is a minor detail, certain problems may require the use of other rather then the other.