# Tagged Questions

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### Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers

Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$. For example: $32 = 21 + 8+3 = F_8+F_6+F_4$ Research ...
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### how to find nth term of different fibonacci series with golden ratio [duplicate]

what i know : if i want to find $Nth$ term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find $6th$ term we use golden ratio ...
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### Fibonacci number “alpha-beta” induction proof [duplicate]

The Question: Use mathematical induction to prove the following: let n be a positive integer and let alpha = [1+sqrt(5)]/2 and beta = [1-sqrt(5)]/2. Then the nth Fibonacci number f_n is given by ...
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### Fibonacci… Easier by induction or directly via Binet's formula

I have tried both for several of them and haven't been able to get anywhere in 3 hours of work. It seems to not matter which method I choose, I end up in the middle of a HUGE mess of algebra. Could ...
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### Fibonacci induction stuck in adding functions together

Using Fibonacci... I am Proving: $$f_3 + f_6 + \cdots + f_{3n} = \frac12(f_{3n+2}-1)$$ I did the assumption of $f_1$ which gave $\mathrm{LHS}=2=\mathrm{RHS}$. For the second part where it is $n+1$ ...
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### how to find nth term in a fibonacci series or sum of a series of fibonacci numbers

A series is given as $1,6,7,13,20,33,.......$ and so on Find the sum of first 52 terms? what i know is The sum of the first n Fibonacci numbers is the [(n + 2)nd Fibonacci number - 1] . So the sum ...
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### Prove the following identity for Fibonacci numbers

Prove this: for any positive integer $a,b,c$, $F_{a+b+c+3}=F_{a+2}(F_{b+2}F_{c+1}+F_{b+1}F_c)+F_{a+1}(F_{b+1}F_{c+1}+F_bF_c)$ Is there any way other than induction to prove this?
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### Show that $f(2n)= f(n+1)^2 - f(n-1)^2$

Let $f(n)=f(n-1)+f(n-2)$ be the Fibonacci sequence with $f(0)=0,f(1)=1$. Show that $$f(2n)= f(n+1)^2 - f(n-1)^2.$$ I have tried several different approaches to this problem. Both inducting from the ...
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### Prove the given property of the Fibonacci numbers directly

The definition of the Fibonacci numbers is as follows: $F(0)=0$, $F(1)=1$, $F(n)=F(n-2)+F(n-1)$ for $n ≥ 2$. Prove the given property of the Fibonacci numbers directly from the definition (hint: do ...
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### Prove that $F_{n}^{2}+F_{n+1}^{2}=F_{2n+1}$ [duplicate]

Prove that $F_{n}^{2}+F_{n+1}^{2}=F_{2n+1}$ This identity holds for $n>=1$ Instead of using induction, how do I prove it in a geometry approach?
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### Prove $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$

Prove the identity: $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$, where $F_i$ denotes a Fibonacci number. How can I prove it using a geometric approach?
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### Fibonacci sequence: how to prove that $\alpha^n=\alpha\cdot F_n + F_{n-1}$?

Let $F_n$ be the $n$th Fibonacci number. Let $\alpha = \frac{1+\sqrt5}2$ and $\beta =\frac{1-\sqrt5}2$. How to prove that $\alpha^n=\alpha\cdot F_n + F_{n-1}$? I'm completely stuck on this question. ...
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### “Fat” sets of integers and Fibonacci numbers

Let us call a set of integers "fat" if each of its elements is at least as large as its cardinality. For example, the set $\{10,4,5\}$ is fat, $\{1,562,13,2\}$ is not. Define $f(n)$ to count the ...
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### Verify the following identity for Fibonacci numbers

This is a homework problem that I would very much appreciate some help with. Thanks!
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### recurrence and fibonacci [closed]

could someone possibly help me with a proof. prove $a_n = F_{2n-1}$ for fibonacci numbers and a recurrence relation where $a_1 = 1$ $a_2 = 2$ $a_3 = 5$ $a_4 = 13$ $a_5 = 34$ 89,233,610,1597 ...
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### Recurrence relation, Fibonacci numbers

$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$. $(i)$ Assume that all $a_n$ are integers. Prove that they are all odd and the integers $a_n$ and $a_{n+1}$ ...
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### Another Bijective proof for Fibonacci Identities

I'm going through a past exam, and this question popped up: Prove: $3P_n = P_{n-2} + P_{n+2},\,n>2$ Where $\tilde{P}_j = \{\textrm{monomer-dimer pavings on a } 1\times j \textrm{ board}\}$ and ...
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### How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$

I am trying to prove it by induction, but I'm stuck $$\mathrm{fib}(0) = 0 < 0! = 1;$$ $$\mathrm{fib}(1) = 1 = 1! = 1;$$ Base case n = 2, $$\mathrm{fib}(2) = 1 < 2! = 2;$$ Inductive case ...
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### Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
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### How to find the closed form to the fibonacci numbers? [duplicate]

Possible Duplicate: Prove this formula for the Fibonacci Sequence How to find the closed form to the fibonacci numbers? I have seen is possible calculate the fibonacci numbers without ...
Possible Duplicate: Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer This is a homework question so I'm looking to just be nudged in ...