5. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{4}, a_{2}=\frac{1}{5}$ and $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n} a_{n+1}=n a_{1} a_{n+1}$ for any positive integer $n$, then $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{97}}$ is $\qquad$ .
5. 5044 Explanation: From $a_{1} a_{2}+a_{2} a_{3}=2 a_{1} a_{3}$ and $a_{1}=\frac{1}{4}, a_{2}=\frac{1}{5}$ we get $a_{3}=\frac{1}{6}$, from $a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}=$ $3 a_{1} a_{4}$ we can get $a_{4}=\frac{1}{7}$, and further it can be proven by mathematical induction that $a_{n}=\frac{1}{n+3}$, thus $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{97}}=4+5+\cdots+$ $100=5044$.