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Prove A<B

Albert

Well-known member
Jan 25, 2013
1,225
A=$1001^{999}$

B=$1000^{1000}$

Prove :A<B

(note :any calculation tools are not allowed ,also no use of log function)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: prove A<B

I would use the binomial theorem to write:

\(\displaystyle 1001^{999}=(1000+1)^{999}=\sum_{k=0}^{999}{999 \choose k}1000^{999-k}\)

Since we have 1000 terms, and each term would have to be equal to $1000^{999}$ in order for the summation to be equal to $1000^{1000}$, yet for $0<k$ we find:

\(\displaystyle {999 \choose k}<1000^k\)

we may therefore conclude that:

\(\displaystyle 1001^{999}<1000^{1000}\)
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Re: prove A<B

A=$1001^{999}$

B=$1000^{1000}$

Prove :A<B

(note :any calculation tools are not allowed ,also no use of log function)
Are we allowed to use the fact that $\bigl(1+\frac1n\bigr)^n$ increases to $e$ as $n\to\infty$? If so, then $\bigl(1+\frac1n\bigr)^n < n+1$ whenever $n\geqslant2$. It follows that $(n+1)^n < (n+1)n^n$ and therefore $(n+1)^{n-1} < n^n.$
 

mathworker

Active member
May 31, 2013
118
Re: prove A&lt;B

I would use the binomial theorem to write:

\(\displaystyle 1001^{999}=(1000+1)^{999}=\sum_{k=0}^{999}{999 \choose k}1000^{999-k}\)

Since we have 1000 terms, and each term would have to be equal to $1000^{999}$ in order for the summation to be equal to $1000^{1000}$, yet for $0<k$ we find:

\(\displaystyle {999 \choose k}<1000^k\)

we may therefore conclude that:

\(\displaystyle 1001^{999}<1000^{1000}\)
as you are using \(\displaystyle 1000^k\)under sigma i think you can't compare \(\displaystyle {999 \choose k}\) with
\(\displaystyle 1000^k\).....:confused:

- - - Updated - - -

is it correct to use \(\displaystyle 1000^K\) under sigma
 

Albert

Well-known member
Jan 25, 2013
1,225
Re: prove A<B

I like Opalg 's method (my solution is similar to his)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: prove A&lt;B

as you are using \(\displaystyle 1000^k\)under sigma i think you can't compare \(\displaystyle {999 \choose k}\) with
\(\displaystyle 1000^k\).....:confused:

- - - Updated - - -

is it correct to use \(\displaystyle 1000^K\) under sigma
My intended purpose was to recognize that for $0<k$, we have:

\(\displaystyle {999 \choose k}1000^{999-k}<1000^{999}\)

and since:

\(\displaystyle \sum_{k=0}^{999}1000^{999}=1000\cdot1000^{999}=1000^{1000}\)

we must therefore have:

\(\displaystyle \sum_{k=0}^{999}{999 \choose k}1000^{999-k}=1001^{999}<1000^{1000}\)
 

mathworker

Active member
May 31, 2013
118
Re: prove A&lt;B

My intended purpose was to recognize that for $0<k$, we have:

\(\displaystyle {999 \choose k}1000^{999-k}<1000^{999}\)

and since:

\(\displaystyle \sum_{k=0}^{999}1000^{999}=1000\cdot1000^{999}=1000^{1000}\)

we must therefore have:

\(\displaystyle \sum_{k=0}^{999}{999 \choose k}1000^{999-k}=1001^{999}<1000^{1000}\)
okay i got it, you are comparing each term to \(\displaystyle 1000^999\)....THANK YOU