The Fibonacci sequence is NOT a geometric sequence.
Suppose the Fibonacci sequence is a geometric sequence (since the differences of the first few terms repeats the sequence), how to show the proof that it has a common ratio that would give an approximate solution to find the nth term of the sequence?
I want to have an idea how the proof goes.
The Fibonacci sequence obeys to the difference equation...
... with the 'initial conditions' and . The general solution of (1) is...
... and the constants and are found from the 'initial constants'. Properly speacking the Fibonacci sequence is the sum of two geometric sequences...
I can't find a way to use Geometric Sequences on this problem.
So I solved the old-fashioned way.
Suppose the Fibonacci sequence is a geometric sequence.
How to prove that it has a common ratio that would give an approximate
. . solution to find the term of the sequence?
We have this sequence:
. . where: .
. . Divide by
. . Quadratic Formula: .
We know that: .
. . Solve the system: .
. . . . . .
Note that: .
. . . . . . . . .where = the Golden Ratio.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
The approximations are quite close!
Thank you! My paper has shown me a bit of this but i don't really get it because it stopped at the middle and assumed that i know the conclusion. Thank you very much, this will help me understand my paper.
Thank you! Do you mean i have to read a book on differential equations? What is difference equation anyway? Is that different from differential equations? Well, i believe, the theory of recurrence relations will help.
Thank you very much! God bless!!
Difference equations are different from differential equations, but a lot of the solution methods are analogous. You even have an analogy of the Laplace transform in the Z transform.
There's even another, tighter, relationship between the two: if you discretize a differential equation, you get a difference equation.
Recurrence relation - Wikipedia, the free encyclopedia
Effectively the precise general term is recurrence relation , even if the term difference equation is more 'traditional' and widely used again...