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

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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## closed as off-topic by 6005, Mathmo123, Grigory M, Adam Hughes, quidJan 7 '15 at 22:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – 6005, Mathmo123, Grigory M, Adam Hughes, quid
If this question can be reworded to fit the rules in the help center, please edit the question.

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

## 1 Answer

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