query: Proof of sum of ratio of Fibonacci numbers is less than 2
positive_passages.docid: 146826
positive_passages.text: Fibonacci Sequence and series limits
negative_passages.docid: 64328
negative_passages.text: How to prove that the Binet formula gives the terms of the Fibonacci Sequence?
negative_passages.docid: 68774
negative_passages.text: Proving that $f(x)$ is less than or equal to $1+\pi/4$
negative_passages.docid: 88907
negative_passages.text: Solving Fibonaccis Term Using Golden Ratio ConvergEnce
negative_passages.docid: 39739
negative_passages.text: Fibonacci identity: $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$
negative_passages.docid: 30672
negative_passages.text: Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
negative_passages.docid: 120976
negative_passages.text: Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
negative_passages.docid: 137215
negative_passages.text: Cesaro identity for Fibonacci numbers: $ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k$
negative_passages.docid: 6366
negative_passages.text: Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 89576
negative_passages.text: Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$
negative_passages.docid: 14784
negative_passages.text: Inductive proof of the closed formula for the Fibonacci sequence
negative_passages.docid: 119891
negative_passages.text: Divisibility and the Fibonacci sequence
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: 45392
negative_passages.text: Every natural number can be written as the sum of distinct Fibonacci numbers?
negative_passages.docid: 42435
negative_passages.text: Proof of this result related to Fibonacci numbers: $\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$?
negative_passages.docid: 48610
negative_passages.text: Fibonacci Numbers Proof: $ f_n = \binom n0 + \binom{n-1}1 +\dots+ \binom{n-k}k$