$t=\frac{30^{65}-29^{65}}{30^{64}-29^{64}}$, find the closest pair of integers, a and b, such that, $a \lt t \lt b$. $t=\frac{30^{65}-29^{65}}{30^{64}-29^{64}}$
find the closest pair of integers, a and b, such that, $a \lt t \lt b$.
$30=1+29$
$(1+29)^{65}=(1+29)(1+29)^{64}$

$t=30+(29^{64}/d)$ with $d=30^{64}-29^{64}$. Since $30^{64}=29^{64}(1+1/29)^{64}\gt29^{64}\,(1+64/29)$, $d\gt29^{64}\cdot(64/29)$ and $30\lt t\lt30+(29/64)\lt31$.
Likewise, $s=((a+1)^{n+1}-a^{n+1})/((a+1)^n-a^n)$ yields $a+1\lt s\lt a+1+(a/n)$ hence $a+1\lt s\lt a+2$ as long as $a\lt n$.
