query: If $a_n$ is the $n$th term of the Fibonacci sequence, then prove that $a_{n+1}^2 - a_n a_{n+2} = (-1)^n$.
positive_passages.docid: 155995
positive_passages.text: Fibonacci proof question: $f_{n+1}f_{n-1}-f_n^2=(-1)^n$
negative_passages.docid: 17902
negative_passages.text: Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$.
negative_passages.docid: 105882
negative_passages.text: Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
negative_passages.docid: 100375
negative_passages.text: Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
negative_passages.docid: 144890
negative_passages.text: Recurrence $f_{n+2}=af_{n+1}+bf_n$
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: 34025
negative_passages.text: Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
negative_passages.docid: 100781
negative_passages.text: Prove the $n$th Fibonacci number is the integer closest to $\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n$
negative_passages.docid: 25583
negative_passages.text: Show that the sequence $a_0 = 1$, $a_{n+1 }= \sqrt{2+a_n}$ is monotonically increasing
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
negative_passages.docid: 33011
negative_passages.text: Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
negative_passages.docid: 111353
negative_passages.text: Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
negative_passages.docid: 8814
negative_passages.text: Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy
negative_passages.docid: 124917
negative_passages.text: Fibonacci numbers and proof by induction
negative_passages.docid: 21137
negative_passages.text: Recurrence relation, induction and Fibonacci numbers
negative_passages.docid: 97656
negative_passages.text: Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$