Which term is bigger? $\sqrt[102]{101}$ or $\sqrt[100]{100}$ Which term is bigger? $\sqrt[102]{101}$ or $\sqrt[100]{100}$
I tried AM-GM but didn't succeed. 

$101^{50} = \sum_{k=0}^{50} \binom{50}{k} 100^{50-k}$.
${101^{50} \over 100^{51} } = {1 \over 100}  \sum_{k=0}^{50} \binom{50}{k} {1 \over 100^k} = {1 \over 100} (1+ {1 \over 100})^{50} = {1 \over 100} (1+ {{1 \over 2} \over 50})^{50} \le {1 \over 100} \sqrt{e} <1$.
