# Tagged Questions

Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

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### Fibonacci Numbers and Legendre symbol

How to prove congruence below ? $$F_{p-\left( \frac{5}{p}\right)} \equiv 0 \pmod p$$ Where $\displaystyle \left( \frac{}{}\right)$ is legendre symbol, and $\displaystyle p$ is a prime number.
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### Prove fibonacci with matrixes [duplicate]

I have a question which i could not figure out the answer to, it was the hardest of them all that i got and i couldnt figure it out, its a proof of fibonaccis serie using matrixes and i need som help ...
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### Proof with Fibonacci Sequence

I was working on my homework assignment for one of my classes and I have come across a proof question that my classmates and I are finding difficult to answer. The problem is asking us to prove that ...
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### How to show this Fibonacci identity? $f_{3n}=f^3_{n+1} + f^3_n - f^3_{n-1}$

I already know that $f_{n+m}=f_{n-1}f_m + f_nf_{m+1}$. By letting $m=n$ it immediately follows that $f_{2n}=f_{n}(f_{n+1} + f_{n-1})$ and from that we get $f_{2n}=f^2_{n+1} - f^2_{n-1}$. From this ...
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### Let $F_n$ denote the nth Fibonacci number and prove that the following re true for every possible integer $n$

$$\sum_{i = 1}^n F_{i}^2 = F_n F_{n+1}$$ -I solved a similar Fibonacci sequence that was the following: $$\sum_{i = 1}^n F_i = F_{n + 2} - 1$$ But, I am having trouble with this one, any help is ...
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### Prove that $\sum^{n}_{i=0}\binom{n}{i}F_{i}=F_{2n}$ [duplicate]

I am asked: Let $F_{i}$ denote the $i$-th Fibonacci number. Prove that $$\sum^{n}_{i=0}\binom{n}{i}F_{i}=F_{2n}$$ I have the base case and the inductive hypothesis, but I'm not sure what ...
### How to prove this series about Fibonacci number: $\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$? [duplicate]
How to prove this series: I have no idea where to start. $$\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$$ where $F_{1}=1,~F_{2}=1,~F_n=F_{n-1}+F_{n-2},~~n\geq 3$.