Tagged Questions
4
votes
2answers
94 views
Prove $((n+1)!)^n < 2!\cdot4!\cdots(2n)!$
so I know I need to prove this via induction, but I am somewhat stuck. Here is what I have does so far.
Let $p(n) = (n+1)!^n \le 2!\cdot4!\cdot\ldots\cdot(2n)!$
$p(2) = 3!^2\le 2!\cdot4!$
Assume ...
2
votes
3answers
202 views
Sum of reciprocals of factorials
Could you help me count this sum:
$$ \sum_{n=1}^{9} \frac{1}{n!} $$
I don't think I can use binomial coefficients.
2
votes
3answers
117 views
Algebraic manipulation of binomial theorem
Prove, by algebraic manipulation, that:
\[ {{2n} \choose {n}} + {{2n} \choose {n+1}}={1\over2} {{2n+2} \choose {n+1}} \]
11
votes
1answer
395 views
Factorial canceling on expansion of binomial coefficients on Concrete Mathematics
On Concrete Mathematics section 5.5, which is teaching the hypergeometric functions, generalized factorials is defined as:
\[
\frac 1 {z!} = \lim_{n \to \infty} \binom{n+z}{n}n^{-z}
\]
where
\[
...
0
votes
2answers
83 views
What is the minimum number of friends Sally can have if she can invite a different group of friends to her house every night for a year?
The question: Sally has $N$ friends and likes to invite them over in small groups for dinner. She calculated that she can invite a different group of $3$ friends to dinner at her house every night for ...
3
votes
0answers
152 views
How to find $\beta$ and $\alpha$?
$\mathbb{P}$ is the prime numbers set.
$p \in \mathbb{P}$
$a,b,c \in \mathbb{N}$
$n=a p^b+c$ where
$c= n\bmod p$
$b$ is the highest power of $p$ who divides $n-c$
How to find $\beta$ where ...
2
votes
1answer
240 views
Use of algebra and factorials for a question related to proof by induction
$$
\begin{align*}
&= (n+1)! − 1 + ( (n+1) · (n+1)! )\\
&= (n+1)! (1+n+1) − 1\\
&= (n+1)! (n+2) − 1\\
&= (n+2)! − 1\\
\end{align*}
$$
I'm confused at how the first ...
