This question already has an answer here: Inductive proof of the closed formula for the Fibonacci sequence [duplicate] 2 answers

I saw this pattern of binary numbers with constraints first number should be 1 , and two 1’s cannot be side by side. Now as an example 1 = 1 10 = 1 100,101 = 2 1000,1001,1010 = 3 10000,10001, 10010, 10100, 10101 = 5 Strangely I see the numbers we can form of this […]

Could anyone help me with this problem? $$ \sum_{j=0}^{n}\binom{n}{j}F_{n+1-j}= F_{2n+1} $$ I used induction and was able to get to this: $$ 2\sum_{j=0}^{n}\binom{n}{j}F_{n-j}= F_{2n} $$ However I still do not know how to prove the second equality. Really appreciate any help

If $F_n$ denotes the $n$-th Fibonacci number ($F_0 = 0, F_1 = 1, F_{n+2} = F_{n+1} + F_n$), show that $5^n$ divides $F_{5^n}$.

Given the recursive equation : $$F_n+F_{n-1}+⋯+F_0=3^n , n\geq0$$ A fast solution that I can think of is placing $n-1$ instead of $n$ , and then we’ll get : $$F_{n-1}+F_{n-2}+⋯+F_0=3^{n-1} $$ Now subtracting both equations : $$F_n+F_{n-1}+⋯+F_0 – (F_{n-1}+F_{n-2}+⋯+F_0) = 3^n – 3^{n-1} $$ $$ F_n = 3^n – 3^{n-1}$$ But how can I do that […]

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each proﬁnite integer $s$, one can in a natural way deﬁne the $s$th Fibonacci number $F_s$, which is itself a proﬁnite integer. Namely, given $s$, one can choose a sequence of positive integers $n_1, n_2, n_3,\dots$ that […]

Let’s define the Fibonacci numbers as $F_0=1$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. Using this recurrence I was able to calculate the generating function of the Fibonacci numbers to be $-\frac{1}{x^2+x-1}$. Now, it can be proved that $F_n$ counts the list of $1,2$ with sum $n$. Is there a way to find the generating function using this model […]

I was looking at the decomposition of Fibonacci numbers using the definition of $F_n = F_{n-1} + F_{n-2}$, and noted the pattern in the coefficients of the terms were Fibonacci numbers. It appears to hold, and I believe it’s true, but I haven’t seen a proof for it. Does one exist? $F_n = 1F_{n-1} + […]

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 effort: The base step it’s simple: Let $k=1$ it can be britten as $k=1=F_2$ For the inductive step I considered: Let $k = k F_2 […]

I have the following theorem to prove by induction: $$ F_{n} \cdot F_{n+1} – F_{n-2}\cdot F_{n-1}=F_{2n-1} $$ It is mentioned in my script that the proof should be possible only by using the recursion of Fibonacci numbers:$$F_{n+1}=F_{n}+F_{n-1}$$ I was only able to proof this theorem by using other theorems, as I was not able to […]

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