Estimate of a combination number by another

Task number: 3854

Estimate \(\binom{2m+1}{m}\) from both sides by a function of \(\binom{2m}{m}\).

  • Solution

    From Pascal’s triangle we have \(\binom{2m+1}{m}=\binom{2m}{m}+\binom{2m}{m+1}\), while \(\binom{2m}{m}>\binom{2m}{m+1}>0\).

    Then straightforwardly.

  • Answer

    The desired estimate could be e.g. \(\binom{2m}{m}<\binom{2m+1}{m}<2\binom{2m}{m}\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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