Question: Prove that $n^n&gt;\left(\dfrac{n+1}{2}\right)^{n+1}$ for all positive integer $n&gt;1$.     I could not understand what should be the initial approach.
121264
I want to prove that for each real number $x&gt;0$, the statement  $${\left(\frac{x+1 } {2}\right)}^{\!x+1} \leq x^x$$  is correct.   I’ve tried using mathematical induction methods, but haven’t found a solution. Although, using any method is appreciated.
33562
I can show that for $x &gt; 0$ and $r_{i} &gt; 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ However, I can't do this using straight up induction, strong or weak. Can someone do this?   You can show the inequality holds for $n$ if it holds for $n+1$ using $r_{n+1} = (r_1 \cdots r_n)^{1/n}$, so going down isn't a problem, but going up has eluded me. Additionally, you can show that if it holds for $n=a$ and $n=b$ then it holds for $n=ab$. Using powers of two and the above is sort of a way to prove the above by induction but it certainly isn't "normal".  You can also prove the above using Lagrange multipliers (which isn't surprising). The last proof I know is where you compare coefficients of $x^k$ and use the AM-GM inequality.
3821
Can you help me with this limit? What do I have to do? I'm lost.  $$\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$$  The solution given is $\dfrac{1}{2}$.
63335
Prove that $1&lt;\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n+1}$.  By using the Mathematical induction. Suppose the statement holds for $n=k$.  Then for $n=k+1$. We have $\dfrac{1}{k+2}+\dfrac{1}{k+3}+...+\dfrac{1}{3k+1}+\dfrac{1}{3k+2}+\dfrac{1}{3k+3}+\dfrac{1}{3k+4}=(\dfrac{1}{k+1}+\dfrac{1}{k+2}+\dfrac{1}{k+3}+...+\dfrac{1}{3k+1})+(\dfrac{1}{3k+2}+\dfrac{1}{3k+3}+\dfrac{1}{3k+4}-\dfrac{1}{k+1})$  we know $\dfrac{1}{k+1}+\dfrac{1}{k+2}+\dfrac{1}{k+3}+...+\dfrac{1}{3k+1}&gt;1$  What can we do for $(\dfrac{1}{3k+2}+\dfrac{1}{3k+3}+\dfrac{1}{3k+4}-\dfrac{1}{k+1})$?
14779
How would one prove that  $$(n+1)^{n-1}&lt;n^n \ \forall n&gt;1$$  I have tried several methods such as induction.
119998
I need help proving $a_n=\left(\dfrac{n+1}{n}\right)^n$ is increasing sequence on the positive integers. An exercise in the analysis book by Mattuck asks to prove $a_n=\left(\dfrac{2^n+1}{2^n}\right)^{2^n}$ is increasing. But this is easy since $\left(\dfrac{2^n+1}{2^n}\right)^{2^n}=\left(\left(\dfrac{2^n+1}{2^n}\right)^{2}\right)^{2^{n-1}} =\left(\dfrac{2^{2n}+2\left(2^n\right)+1}{2^{2n}}\right)^{2^{n-1}}=$ $\left(\dfrac{2^{n-1}+1+\frac{1}{2^{n+1}}}{2^{n-1}}\right)^{2^{n-1}}&gt;\left(\dfrac{2^{n-1}+1}{2^{n-1}}\right)^{2^{n-1}}$ as desired. I am having difficulty in the general case: I must prove $\left(\frac{n+2}{n+1}\right)^{n+1}&gt;\left(\frac{n+1}{n}\right)^n$. This is equivalent to $(n+2)^{n+1}n^n&gt;(n+1)^{2n+1}$ which is the same as proving $\frac{n+2}{n+1}&gt;\left(\frac{(n+2)(n)}{(n+1)^2}\right)^n$ which is the same as $\frac{n+2}{n+1}&gt;\left(1-\frac{1}{(n+1)^2}\right)^n$ I'm stumped. Thanks in advance. Oh, and I don't want to use the derivative.
114967
Prove by induction that $$(a_{1}+a_{2}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right)\geq n^{2}$$  where $n$ is a positive integer and $a_1, a_2,\dots, a_n$ are real positive numbers   Hence, show that  $$\csc^{2}\theta +\sec^{2}\theta +\cot^{2}\theta \geq 9\cos^{2}\theta$$  Please help me.  Thank you!
110000
How can I prove that $n!&lt;n^n$ for every $n&gt;1$ using the induction principle?  If I put $n=2$ I get $2&lt;4$ so I know that $n!&lt;n^n$ is true.  Now I don't know how to prove that $(n+1)!&lt;(n+1)^{n+1}$. I can write $(n+1)!=n!(n+1)$ but then I don't know how to go on.
86548
Prove that: $$6 \not\left|\ \left\lfloor\frac 1 {(\sqrt[3]{28} - 3)^{n}}\right\rfloor \ (n \in Z^+)\right.$$ ($\lfloor x\rfloor$ = largest integer not exceeding $x$)  I am very bad as English and number theory, please help me
94373
Prove that $n$ is even   $$\left(\dfrac {\cos A + \cos B}{\sin A-\sin B}\right)^n + \left(\dfrac {\sin A+\sin B}{\cos A- \cos B}\right)^n = 2\cot^n \left(\dfrac {A-B}{2}\right)$$  Please help. I didn't get any idea regarding this.
93723
For every positive integer $d$, we let $\tau\left(d\right)$ be the number of positive divisors of $d$. Prove that \begin{align} \sum_{d|n} \tau^3(d) = \left(\sum_{d|n} \tau (d)\right)^2 \end{align} for each positive integer $n$, where the sums range over all positive divisors $d$ of $n$.  Now I only know that both sides are multiplicative arithmetic functions in $n$. Could you tell me what I need to do next?
2770
Show that if $n&gt;2$, then $(n!)^2&gt;n^n$.     My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n&gt;(n+1)^{n-1}$ Here, I tried to use induction again without any luck. I also took log of both sides, but I did not get anything. Please help!
26124
Can someone look at my proof. I am supposed to prove by induction. The question is to prove the following:  $$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right)  }{ 2 } .$$  If $n=1$  Then  $$1=\sum_{i=1}^1{i}=\frac{1(1+1)}{2}=1$$  Now assume $n=k$.  Thus  $$\sum_{i=1}^k{i}=\frac{k(k+1)}{2}=1$$  If now $n=k+1$  $$\sum_{i=1}^{k+1}{i}=\sum_{i=1}^k{i}+(k+1)= \frac{(k+1)(k+2)}{2}$$  Is there anything more that I have to do, or is this it?
101831
I am trying to prove that for every element of $\mathbb{N}$, that                    $2^n \leq 2^{n+1} - 2^{n-1} - 1.$ I started by showing that initial case, of $n=1$, is true. Then I proceed to the inductive step by assuming that for some $k$ in $\mathbb{N}$,                     $2^k \leq 2^{k+1} - 2^{k-1} - 1$.  Then, for $n=k+1$,                    $2^{k+1} \leq 2^{k+2} - 2^{k} - 1$. Is this sufficient to prove the statement by induction or do I have to expand the exponentials to rearrange the inequality in a different way?
108873
I cannot prove that the function $$f (n) = \left(1 + \frac1n\right) ^ {n + 1},$$ defined for every positive integer $n$, is strictly decreasing in $n$. I already tried to prove by induction and also tried to prove by calculating the difference between $f (n + 1)$ and $f (n)$. I need help.
18138
I would like to prove that  $$a^{2^n}\equiv 1 \pmod {2^{n+2}}$$  I tried induction but could not get it.  Thank you very much!
119945
I'm about to prove that for any $a,b&gt;0$ and $n\in\mathbb{N},$ the inequality: $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$ holds.   By induction I get: $$\left(\frac{a+b}{2}\right)\cdot\frac{a^n+b^n}{2}$$ $$=\frac{1}{2}\cdot\frac{a^{n+1}+b^{n+1}+ab^n+ba^n}{2}$$ $$=\frac{1}{2}\left(\frac{a^{n+1}+b^{n+1}}{2}+\frac{ab^n+ba^n}{2}\right).$$  Now I have to prove that $$\frac{1}{2}\left(\frac{a^{n+1}+b^{n+1}}{2}+\frac{ab^n+ba^n}{2}\right)$$ $$\leq\frac{a^{n+1}+b^{n+1}}{2}$$ $$\Leftrightarrow a^{n+1}+b^{n+1}\geq ab^n +ba^n.$$ But I don't know how to prove that one, is it possible to be proved by induction ? Thank you.