Prove that $1\cdot 1! + 2\cdot 2! +\dots+n\cdot n! = (n + 1)! - 1$
57780
$\sum_{i=1}^n i\cdot i! = (n+1)!-1$ By Induction
63553
Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$
148458
$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
91933
Evaluate $\sum_{k=1}^nk\cdot k!$
33903
Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
34299
Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
5257
Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2
7302
$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?
2675
Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
86900
Formula for the $1\cdot 2 + 2\cdot 3 + 3\cdot 4+\ldots + n\cdot (n+1)$ sum
152588
Expression for $n+n(n-1)+n(n-1)(n-2)+...+n!$
111298
Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
87411
Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
56624
Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
88995
Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
98946
Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$
14986
Proving a combinatorics equality: $\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}$