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Prove that for $n>1$, sequence $${n \choose 0},{n \choose 1},{n \choose 2},...,{n \choose \lfloor n/2\rfloor}$$ is strictly monotonically increasing.

If I will show, that $${n \choose \lfloor n/2\rfloor}-{n \choose \lfloor n/2-1\rfloor} >0$$ will that be the end of proof?

matematiccc
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1 Answers1

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Hint: Prove that $\dfrac{n\choose k}{n\choose k+1} < 1$. There'll lots of cancellation.

lhf
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  • Ok. I proved it. However I didn't use assume that $0\leq k<\lfloor n/2\rfloor-1$. Is it the end of proof? – matematiccc Nov 10 '18 at 18:47
  • Because for $k=\lfloor n/2\rfloor-1$, ${n\choose k} \le {n\choose k+1}$, and the sequence must be strictly monotonically increasing. Sorry, I've forgotten to write this. – matematiccc Nov 10 '18 at 19:13
  • Why is $k=\left \lfloor{n\over 2}\right \rfloor-1$ excluded? – miniparser Nov 10 '18 at 19:14