Expert: Josh - 2/13/2005

Question
Hello:

I want to thank you for the reply.  I noticed that you started with the second number "1" from the sequence when you refer to these numbers as in 1 is T[1], 2 is T[2], 3 is T[3], 5 is T[4], 8 is T[5], 13 is T[6], etc.
Why is the first 1 in parentheses, as in (1), 1, 2, 3, ..., not regarded as the first number in the sequence?
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Followup To
Question -
Hello:

Can you explain how the percentages of 62 and 38 are determined from the following?

"The Fibonacci number sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, ...) is constructed by adding the first two numbers to arrive at the third.  The ratio of any number to the next larger number is 62%, The inverse of 62% is 38%."

I thank you for your reply.
Answer -
Hello,

The ratio is actually 62% or round about.
If we let T[n] be the nth term in the Fibonacci sequence and define the initial conditions, such that
T[0]=1 and T[1]=1, then, T[n]=T[n-1]+T[n-2] for n>=2.

It claims that T[n]/T[n-1] is 1.62 or round about.
In the case of n=6, T[6]=13,T[5]=8, we get T[6]/T[5]=13/8=1.625. Roughly speaking, the T[6] term is about 62% larger than the previous term, T[5].

The "38%" is called the complement of "62%" simply because 38%=100%-62%.

Cheers.

Answer
It doesn't matter what the index of the first term is. You can begin with i=1 instead of i=0 if you like.

The important thing is that we specify the initial conditions (i.e., give the values of the first two terms).

If it helps, you can let T[1]=1, T[2]=1, then, T[i]=T[i-1]+T[i-2] for i>=3.

You can also let T[-2]=1,T[-1]=1 and define the recurrent relation by T[i]=T[i-1]+T[i-2] for i>=0. In this case, T[j] is undefined for j<-2.

But it would be nice if T[j] is undefined whenever j is negative. That's why I prefer to follow the convention
T[0]=1,T[1]=1 etc.