Linked Questions

4
votes
4answers
129 views

Greatest of the numbers given [duplicate]

To find out the greatest among the number given below: $3^{1/3}, 2^{1/2}, 6^{1/6}, 1, 7^{1/7}$ I have plotted the following graph using graph plotter which is shown below: It can be concluded that $...
3
votes
2answers
102 views

Greatest number in the sequence [duplicate]

I have seen this problem in a book . But I don't know what should be the solution . Question is There is a sequence defined by $$\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4},\cdots,\sqrt[n-1]{n-...
1
vote
3answers
68 views

How to know which value is bigger? [duplicate]

Which is bigger between $2018^{2019}$ or $\ 2019^{2018}\ $? When taking logs of both sides and I get: $2019\log(2018)\ $ and $\ 2018 \log(2019)$ I know $\log 2019\gt \log 2018$ so does this mean ...
17
votes
10answers
9k views

Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no ...
0
votes
5answers
254 views

Prove that $\sqrt[n]{n} > \sqrt[n+1]{n+1}$ without calculus?

I'm stuck with this sample RMO question I came across: Determine the largest number in the infinite sequence $\sqrt[1]{1}$, $\sqrt[2]{2}$, $\sqrt[3]{3}$, ..., $\sqrt[n]{n}$, ... In the solution to ...
-1
votes
3answers
185 views

Compare two below natural numbers: $2016^{2017} < 2017^{2016}$ [closed]

Help me Compare the two following natural numbers below $$2016^{2017} < 2017^{2016}?$$ Many thanks.
3
votes
5answers
255 views

proving that $(n-1)^n>n^{n-1}$ [duplicate]

I want to prove that $(n-1)^n>n^{n-1}$, for $n>4$, $n$ is an integer. So I divided by $n^n$ and got: $(1-\frac{1}{n})^{n}>\frac{1}{n}$ I know that $(1-\frac{1}{n+1})^{n+1}$>$(1-\frac{1}{n}...
2
votes
3answers
250 views

Prove that $\sqrt[8]5 > \sqrt[9]6 > \sqrt[10]7 > \cdots$

Prove that $\sqrt[8]5 > \sqrt[9]6 > \sqrt[10]7 > \cdots$ My friend came up with this and gave this to me as a challenge and I'm totally stuck. I have tried proving this by induction $\root{...
-1
votes
3answers
81 views

Which is greatest among the follwing? [closed]

Which is greatest among $2^{1/2}$ $3^{1/3}$ $4^{1/4}$ $6^{1/6}$ $12^{1/12}$ how to approach this??help.
1
vote
2answers
73 views

Prove that $\sqrt[n]n>\sqrt[n+1]{n+1}$ for all $n \geq 3$

Can someone please prove the following? $$\sqrt[n]n>\sqrt[n+1]{n+1} \quad \text{for all } n\geq 3.$$ I have tried lots of different approaches but none of them has worked. I tried induction and ...