Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
4855
How to show that this binomial sum satisfies the Fibonacci relation?
83738
Prove $(k,n+k)=1$ iff $(k,n)=1$
42105
Fibonacci proof by induction
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
89576
Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$
116518
Proof by induction: $n$th Fibonacci number is at most $ 2^n$
132486
Induction proof on Fibonacci sequence: $F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$
18046
Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
144890
Recurrence $f_{n+2}=af_{n+1}+bf_n$
106502
Prove by induction that for the Fibonacci numbers $F(n)$ with $n \ge 6$, $F(n) \ge 2^{n/2}$
132089
How do I prove/disprove that $F_n^2 + F_{n\pm 1}^2 = F_{2n\pm 1}$, where $F_n$ is the $n$th Fibonacci number?
131400
Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
70128
Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
59612
Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers
27217
For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$