5、Insert $n$ real numbers between 1 and 100, such that these $n+2$ numbers form an increasing geometric sequence. Let the product of these $n+2$ numbers be denoted as $T_{n}$, and let $a_{n}=\lg T_{n}, n \geq 1$. Define $b_{n}=\tan a_{n} \cdot \tan a_{n+1}$, then the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$ is $S_{n}=$ $\qquad$
Solve: $\frac{\tan (n+3)-\tan 3}{\tan 1}-n$
It is easy to get $a_{n}=n+2, b_{n}=\tan (n+2) \cdot \tan (n+3)$, using
$$
\begin{array}{c}
\tan 1=\tan (k+1-k) \Rightarrow \tan k \tan (k+1)=\frac{\tan (k+1)-\tan k}{\tan 1}-1 \\
\Rightarrow S_{n}=\frac{\tan (n+3)-\tan 3}{\tan 1}-n
\end{array}
$$