query: Prove that $n$ factorial is larger than $n$ cube if $n$ is large enough
positive_passages.docid: 27147
positive_passages.text: Proving $n! > n^3$ for all $n > a$
negative_passages.docid: 135614
negative_passages.text: Show that $2^n > n^2$ whenever n>4.
negative_passages.docid: 2197
negative_passages.text: Induction principle for $n!<n^n$
negative_passages.docid: 104855
negative_passages.text: Prove that $N \subset Z(G)$
negative_passages.docid: 55273
negative_passages.text: Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$
negative_passages.docid: 78726
negative_passages.text: Prove via mathematical induction that $4n < 2^n$ for all $ n≥5$.
negative_passages.docid: 114404
negative_passages.text: For which positive integer values of $n$ does $a|mn-b?$
negative_passages.docid: 66358
negative_passages.text: Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$?
negative_passages.docid: 150156
negative_passages.text: Proving that $S_n$ has order $n!$
negative_passages.docid: 21612
negative_passages.text: How to prove this form of $n$?
negative_passages.docid: 8609
negative_passages.text: Prove by induction that $n!>2^n$
negative_passages.docid: 4776
negative_passages.text: Prove that the power set of an $n$-element set contains $2^n$ elements
negative_passages.docid: 119808
negative_passages.text: Proving that $n$-cycle has order $n$
negative_passages.docid: 120309
negative_passages.text: Number Theory Inequality Proof
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$
negative_passages.docid: 43764
negative_passages.text: Prove $P_m X P_n$ is Hamiltonian if and only if at least one of $m,n$ is even
negative_passages.docid: 93657
negative_passages.text: Proving “The sum of $n$ consecutive cubes is equal to the square of the sum of the first $n$ numbers.”