# Proof by induction: $n^{n+1} > (n + 1)^n$, $(1 + x)^n \ge 1 + nx$, other inequalities

I'm struggling around my homework. I hope someone will point me the right direction for solving following examples:

• Prove that $n^{n+1} > (n + 1)^n$ for $n > 2$;

• Prove that $(1 + x)^n \ge 1 + nx$; $x \in\Bbb R$; $n \in\Bbb N$;

• Prove that $(2n)! < 2^{2n}(n!)^2$; $n \ge 1$;

• Prove that $2^n > n$:

Thank you a lot.

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Show us what you did and where you are stuck. –  Chris Gerig Sep 28 '12 at 21:49
So, do you not know what induction is? What do you know about induction? –  Graphth Sep 28 '12 at 21:54

For the first note that the inequality is equivalent to $n>(1+\frac1n)^n$

For the second note that $(1+nx)(1+x)=1+(n+1)x + n x^2\ge 1+(n+1)x$ and use this in an induction proof at least for the case $x\ge -1$. Think about it: Is the claim true at all if we allow very negative $x$? What about $x=-3$ and $n=5$?

For the third: Replace each odd factor $k$ in $(2n)!$ by $k+1$ and extract $2n$ factors of $2$ from the $2n$ (now) even factors.

The third is a very trivial induction.

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Did you mean fourth in that last line? –  Brian M. Scott Sep 28 '12 at 21:53
While I realize the word has a (soft) mathematical meaning, it's kind of mean to call others' problems trivial. –  Snowball Sep 28 '12 at 22:07
It is not mean if it is indeed trivial, which it is. –  copper.hat Sep 28 '12 at 23:38