query: Prove the following identity (Fibonacci numbers)
positive_passages.docid: 25961
positive_passages.text: Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$
negative_passages.docid: 112948
negative_passages.text: Fibonacci addition law $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$
negative_passages.docid: 106183
negative_passages.text: Proof by Induction: Alternating Sum of Fibonacci Numbers
negative_passages.docid: 108688
negative_passages.text: An identity involving products of sums
negative_passages.docid: 91914
negative_passages.text: Prove that for the Fibonacci sequence $(F_n)$, $F_n$ divides $F_{2n}$.
negative_passages.docid: 7003
negative_passages.text: Inductive proof of a formula for Fibonacci numbers
negative_passages.docid: 132486
negative_passages.text: Induction proof on Fibonacci sequence: $F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$
negative_passages.docid: 91068
negative_passages.text: Proving that $f_2+f_4+\cdots+f_{2n}=f_{2n+1}-1$ for Fibonacci numbers by induction
negative_passages.docid: 64378
negative_passages.text: Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
negative_passages.docid: 91889
negative_passages.text: Prove $\phi^n = \phi F_n + \text{(another Fibonacci number)}$ using mathematical induction.
negative_passages.docid: 10875
negative_passages.text: proof of binomial identity involving double sum
negative_passages.docid: 133150
negative_passages.text: Fibonacci combinatorial identity: $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$
negative_passages.docid: 10777
negative_passages.text: Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
negative_passages.docid: 50226
negative_passages.text: Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$
negative_passages.docid: 59612
negative_passages.text: Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers
negative_passages.docid: 92418
negative_passages.text: Prove by induction Fibonacci equality
negative_passages.docid: 55481
negative_passages.text: Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$