query: Showing that the $n$-th Fibonacci number $f_n$ satisfies $f_n < 2^n$
positive_passages.docid: 116518
positive_passages.text: Proof by induction: $n$th Fibonacci number is at most $ 2^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: 6443
negative_passages.text: Is there a proof for this Fibonacci relationship?
negative_passages.docid: 20574
negative_passages.text: Understanding the proof for $\gcd (F_n, F_{n+k}) = 1$ where $F_n$ is a Fermat number
negative_passages.docid: 21137
negative_passages.text: Recurrence relation, induction and Fibonacci numbers
negative_passages.docid: 132089
negative_passages.text: 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?
negative_passages.docid: 75373
negative_passages.text: Show that $2^n>n$ for $n\in \mathbb{N}$
negative_passages.docid: 24407
negative_passages.text: How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$
negative_passages.docid: 62254
negative_passages.text: Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$
negative_passages.docid: 119891
negative_passages.text: Divisibility and the Fibonacci sequence
negative_passages.docid: 139625
negative_passages.text: Proof of a Fibonacci identity
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 48610
negative_passages.text: Fibonacci Numbers Proof: $ f_n = \binom n0 + \binom{n-1}1 +\dots+ \binom{n-k}k$
negative_passages.docid: 75329
negative_passages.text: Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$
negative_passages.docid: 25961
negative_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: 43058
negative_passages.text: If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.
negative_passages.docid: 4776
negative_passages.text: Prove that the power set of an $n$-element set contains $2^n$ elements