Which one is bigger $\sqrt[1023]{1024}$ or $\sqrt[1024]{1023}$

Question

I am really stuck with this one.My friend says that it can be solved by $AM-GM$ but I didn't succes.Any hints?

@ritwiksinha: Not "AM or GM", but the AM-GM inequality. (No, it's not clear to me how that would be helpful here). — hmakholm left over Monica, Jul 11 '16 at 09:25
I thought the question to be http://math.stackexchange.com/questions/1612256/showing-that-for-n-geq-3-the-inequality-n1nnn1-holds — lab bhattacharjee, Jul 11 '16 at 09:26
Why would you use AM-GM here? When $1<x_1<x_2$ and $0<y_1<y_2$ don't you trivially have $$x_1^{y_1}<x_1^{y_2}<x_2^{y_2}?$$ — Jyrki Lahtonen, Jul 11 '16 at 09:26

Arthur · Accepted Answer · 2016-07-11T09:46:36.977

19

Raise both numbers to the power of $1023\cdot 1024$ to get $1024^{1024}$ and $1023^{1023}$. Which one looks bigger now?

Alternatively, pick your fravourite from among the two numbers $\sqrt[1023]{1023}$ or $\sqrt[1024]{1024}$, and compare each of the original two numbers to the one you picked.

edited Jul 11 '16 at 09:46

answered Jul 11 '16 at 09:19

Arthur

199,419

@Roby5 Can you compare $\color{red}{\sqrt[1023]{1024}}$ and $\color{blue}{\sqrt[1023]{1023}}$? Can you compare $\color{blue}{\sqrt[1023]{1023}}$ and $\color{red}{\sqrt[1024]{1023}}$? The two blue numbers are the same. What do the two comparisons made say about the two red numbers? – Arthur Jul 11 '16 at 09:45

score 2 · Answer 2 · answered Jul 11 '16 at 10:48

Note that functions $x\mapsto x^n$ and $x\mapsto n^x$ are increasing functions ($x\in\mathbb R_{>0},\ n\in\mathbb N$). Now we have that $$m< n\implies m^m<n^m<n^n\implies (m^m)^{\frac 1 {mn}}<(n^n)^{\frac 1 {mn}}\implies \sqrt[n]m<\sqrt[m]n.$$ Letting $m=2013,\ n=2014$ answers your question.

Which one is bigger $\sqrt[1023]{1024}$ or $\sqrt[1024]{1023}$

2 Answers2