TITLE: Compare $(5/6)^4$ and $(35/36)^{24}$ without calculation
QUESTION [5 upvotes]: Can anyone provide some hint how to compare $(5/6)^4$ and $(35/36)^{24}$ without calculation? Thanks!
After some transformation this question is equivalent to check if $(5/6)^{20}(7/6)^{24}$ is larger than one or smaller than one.

REPLY [3 votes]: Raise both to the power of $3\over2$. Then we will be comparing
$\left(\frac{5}{6}\right)^6\;\text{ to }\;\left(\frac{35}{36}\right)^{36}$, and we might have heard a thing or two about the sequence $\left(\frac{n-1}{n}\right)^n$, which increases and converges to... well, that's not really important here, just note that it increases.