2
votes
1answer
52 views

Proving a Problem involving Fibonacci numbers

I'm working on proving the problem that states $\text {The sequence}$ {$F_n$} $\text {is defined by the} \ F_1=F_2=1, F_{n+2}=F_{n+1}+F_n \ \text {for} \ n \ge 1.$ $\text {For any natural number m, ...
0
votes
1answer
51 views

Base case in the Binet formula (Proof by strong induction)

I don't understand why the author says that the inductive formula does not apply until $u_3$. The formula does not depend on other terms in the Fibonacci sequence, so I thought I could just prove it ...
1
vote
2answers
57 views

Help with a proof I can't quite

Let $(F_j)^\infty_{j=1}$ be the sequence of Fibonacci numbers. For all $n \in \mathbb{N}$, $\sum\limits_{k=1}^{2n-1}F_kF_{k+1}=(F_{2n})^2$. I handled the base case quite well but couldn't go very ...
1
vote
2answers
155 views

Induction proof with Fibonacci numbers

Prove by induction that for Fibonacci numbers from some index $i > 10$ $1.5^i ≤ f_i ≤ 2^i$ Notice! Because Fibonacci number is a sum of 2 previous Fibonacci numbers, in the induction hypothesis ...
3
votes
2answers
61 views

Gcd of every other Fibonacci number

Let $f_n$ be Fibonacci Sequence. $$gcd(f_{n},f_{n+2})=1,\quad \forall\,n\in\mathbb{N}.$$ Prove Could you help me with this one? I have done the base case, I just can't figure out the inductive step. ...
0
votes
3answers
250 views

Prove the Number of Additions of Fibonacci Number Algorithm

I am studying for a final exam and I'm having trouble with this question: The following recursive algorithm FIB takes as input an integer $n \ge 0$ and returns the $n$-th Fibonacci number $F_n$: ...
2
votes
3answers
640 views

Using induction to prove a result about the Fibonacci sequence

The Fibonacci sequence $F_0, F_1, F_2,...,$ are defined by the rule $$F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$$ Use induction to prove that $F_n\geq2^{0.5n}$ for $n\geq 6$ So far I have done the ...
5
votes
1answer
331 views

What is wrong with my induction proof?

The Fibonacci sequence is defined by $a_1 = 1, a_2 = 1$ and for all $n \ge 2, a_{n+1} = a_n + a_{n-1}$. Thus the sequence begins $$1,1,2,3,5,8,13,21,...$$ prove that for all $n \ge 1, a_n < ...
1
vote
2answers
186 views

Prove $F_{n+1}F_{n-1}-(F_{n})^2=(-1)^n$ without induction

I am asked to pove the statement about fibonacci sequence. The task is from the passage about series and sequences. But the proof seems to need induction way, doesn't it? Prove the statement ...
3
votes
1answer
2k views

Induction Proof: Formula for Sum of n Fibonacci Numbers

I am stuck though on the way to prove this statement of fibonacci numbers by induction : my steps: definition: $F_{0}:=0, F_{1}:=1 $ and $F_{n}:=F_{n-1}+F_{n-2}$ The Hypothesis is: $\sum_{i=0}^{n} ...
3
votes
3answers
295 views

Another way to go about proving Binet's Formula

As I showed in another question of mine, it is easy to prove that $$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$ given $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2$. Now, extending $(1)$ ...
3
votes
3answers
165 views

Prove for Fibonacci numbers: $3\mid f(n) \iff 4\mid n$

Let $f(n)$ be the $n$th Fibonacci number. Prove that $$3\mid f(n) \iff 4\mid n$$ I tried to use induction to prove it but I couldn't continue when I reached $n+1$ case.