query: How is: $(n+1)!(n+1)+(n+1) = (n+2)!$?
positive_passages.docid: 146443
positive_passages.text: Factorials, Simplify the addition and multiplication of factorials.
negative_passages.docid: 7894
negative_passages.text: Does it follow that $(n!)^n$ divide $(n^2)!$
negative_passages.docid: 70611
negative_passages.text: Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$
negative_passages.docid: 118372
negative_passages.text: Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
negative_passages.docid: 83738
negative_passages.text: Prove $(k,n+k)=1$ iff $(k,n)=1$
negative_passages.docid: 4315
negative_passages.text: show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
negative_passages.docid: 63661
negative_passages.text: integers $n$ such that $(n-1)!+1$ is a power of $n$
negative_passages.docid: 145519
negative_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
negative_passages.docid: 34374
negative_passages.text: Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$
negative_passages.docid: 56624
negative_passages.text: Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
negative_passages.docid: 152588
negative_passages.text: Expression for $n+n(n-1)+n(n-1)(n-2)+...+n!$
negative_passages.docid: 143357
negative_passages.text: Comparing $2^n$ to $n!$
negative_passages.docid: 95889
negative_passages.text: Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.
negative_passages.docid: 19588
negative_passages.text: $n!+1$ being a perfect square
negative_passages.docid: 5257
negative_passages.text: Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2
negative_passages.docid: 8609
negative_passages.text: Prove by induction that $n!>2^n$
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$