A few clarifications of Fibonacci series

A

Amardeep

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Mar 31, 2021
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If there is a series or set of numbers like
s = 1,3,5,7,9,11,............
then Fibonacci series is defined as
f(s) = 1,4,9,16,25,36,.......
basically it says
f(s) = xo, (xo+x1), (xo+x1+x2), (xo+x1+x2+x3), (xo+x1+x2+x3+x4) ...........

So if one observes the Fibonacci series it is a sum of all the numbers at a particular position in a series and all the number preceding that number

My questions are as follows
1) Is there a series in which instead of sum we do the multiplication? So for example the series becomes
f1(s) = 1,3,15,105,945, 10395,.......
so series is defined as
f1(s) = xo, (xo*x1), (xo*x1*x2), (xo*x1*x2*x3), (xo*x1*x2*x3*x4)...........
If there is such series then what is it called?

2) In Question 1 we have a variant of the Fibonacci series in which we have multiplication of the numbers of the series instead of the addition. Can that be extended to division and subtraction too? Do such series have names?

3) I have heard that Fibonacci series is used in finance and stock market trading. But apart from these applications is there any real world application of Fibonacci series? Does it occur in nature?
 
Amardeep said:
If there is a series or set of numbers like
s = 1,3,5,7,9,11,............
then Fibonacci series is defined as
f(s) = 1,4,9,16,25,36,.......
basically it says
f(s) = xo, (xo+x1), (xo+x1+x2), (xo+x1+x2+x3), (xo+x1+x2+x3+x4) ...........

So if one observes the Fibonacci series it is a sum of all the numbers at a particular position in a series and all the number preceding that number

My questions are as follows
1) Is there a series in which instead of sum we do the multiplication? So for example the series becomes
f1(s) = 1,3,15,105,945, 10395,.......
so series is defined as
f1(s) = xo, (xo*x1), (xo*x1*x2), (xo*x1*x2*x3), (xo*x1*x2*x3*x4)...........
If there is such series then what is it called?

2) In Question 1 we have a variant of the Fibonacci series in which we have multiplication of the numbers of the series instead of the addition. Can that be extended to division and subtraction too? Do such series have names?

3) I have heard that Fibonacci series is used in finance and stock market trading. But apart from these applications is there any real world application of Fibonacci series? Does it occur in nature?
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How are you defining "Fibonacci series"? Nothing here appears to be about the actual Fibonacci sequence, which is 1, 1, 2, 3, 5, 8, 13, ... . See https://www.mathsisfun.com/numbers/fibonacci-sequence.html (In the first sentence, they use the word "series" in its nontechnical sense.)

You seem to be using the term to mean "series" in general. In fact, the Fibonacci sequence is not even a series at all, as it is not a summation. See https://www.purplemath.com/modules/series.htm

1) A sequence can have any form at all, including the product of successive terms of some given sequence.
Your sequence can be denoted as [MATH]\prod_{i=0}^n x_i[/MATH]. I don't think there is a special name for such a sequence; see https://en.wikipedia.org/wiki/Multiplication#Product_of_a_sequence

2) Most sequences have no special names.

3) Just search and find out: https://en.wikipedia.org/wiki/Fibonacci_number See the section on applications. Mostly, it is a curiosity, of only theoretical interest, and used as an example in teaching about inductive proof.
 
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