Suppose that F1 = 1, F2 = 1, F3 =1, F4 = 3, F5 = 5, and in general Fn = Fn-1 + Fn-2 for n ≥ 3 ( Fn is called the nth Fibonacci number.) Prove that F1 + F2 + F3 +...+ Fn = F(n + 2) - 1

Show that the Fibonaccinumbers satisfy the recurrence relation f_n = 5f_n-4 + 3f_n-5 for n = 5, 6, 7,..., together with the initial f_0 = 0, f_1 = 1, f_2 = 1, f_3 = 2, and f_4 = 3.
See the attachment for the full question.

Application of Mathematical Induction
It is an application of Mathematical Induction in proving the relations of Fibonacci Numbers.
To prove: F 2n+1 - Fn Fn+2 = (-1) n for n > or, = 3.

Application of Mathematical Induction
Application of Mathematical Induction
FibonacciNumbers :- The Fibonaccinumbers are numbers that has the following properties.
If Fn represents the nth Fibonaccinumber,
F1 = 1, F2 =1, F3 =2, F4=3, F5 = 5 etc.
We can find the Fibonaccinumber

First part is to find an expression in terms of n, the results of the formula:
and prove the expression is correct?
Secondly
A Fibonacci sequence is the basis for a superfast calculation trick as follows:
Turn your back and ask someone to write down any two positive integers (vertically and one below the othe

Show that |(frac{f_(n+1)}{f_n}) - phi| = frac{1}{(f_n)(phi^{n+1})}
and lim_{n --> infty} frac{f_{n+1}}{f_n} = phi,
where phi is the Golden Ration and is the unique positive root of phi^2 - phi - 1 = 0.
For some discussion on this question, see http://math.stackexchange.com/questions/106049/another-way-to-go-about-provin

The Fibonacci recurrence is F(0) = F(1) = 1, and F(n) = F(n-1) + F(n-2), for n > 1
The values F(0), F(1), F(2), ... form the sequence of Fibonaccinumbers, in which each number after the first two is the sum of the two previous numbers. Let r = (1+ /2). The constant r is called the golden ratio and its value is about 1.62.

A) How many different trains of white (1x1) and/or red (1x2) rods are there of a given length N? Find at least 2 ways to justify your solution
b) Can you find an equation to express the number of trains of length N that can be formed using white, red and/or green (1x3) rods?

Using
Fn+3 = Fn+2 + Fn+1
show that
Fn+3 + Fn = 2Fn + 2
and
Fn + 3 - Fn = 2Fn + 1
It seems intuitive at first but would appreciate a detailed explanation - how does one identify the terms to be used?
Thank you