2. 102 Real numbers $u, v$ satisfy the relation
$$
\begin{aligned}
\left(u+u^{2}+u^{3}+\cdots+u^{8}\right)+10 u^{9}= & \left(v+v^{2}+v^{3}+\cdots+v^{10}\right)+ \\
& 10 v^{11} \\
= & 8 .
\end{aligned}
$$

Determine which of $u$ and $v$ is larger, and prove your conclusion.
[Solution] From the given relation, we know
$$
u0, v>0$. Therefore,
$$
00, \\
g(0)=-80,
\end{array}
$$

Thus, $u$ and $v$ are the unique roots of the equations $f(x)=0$ and $g(x)=0$ in the interval $(0,1)$, respectively.
From equation (1), we know that $u$ is the root of
$$
f_{1}(x)=(10 x-9)\left(x^{9}-1\right)+x-1=0
$$

in $(0,1)$. Since $f_{1}(0)=8>0$ and $f_{1}\left(\frac{9}{10}\right)=-\frac{1}{10}0$. Therefore, $v$ is in the interval $(u, 1)$, hence $u<v$.