Question 119
Find all sequences of positive integers $\{a_{n}\}_{n=1}^{\infty}$, such that $a_{4}=4$ and the identity
$$
\frac{1}{a_{1} a_{2} a_{3}}+\frac{1}{a_{2} a_{3} a_{4}}+\cdots+\frac{1}{a_{n} a_{n+1} a_{n+2}}=\frac{(n+3) a_{n}}{4 a_{n+1} a_{n+2}}
$$
holds true for every positive integer $n \geq 2$.
$$
\frac{1}{a_{1} a_{2} a_{3}}+\frac{1}{a_{2} a_{3} a_{4}}+\cdots+\frac{1}{a_{n} a_{n+1} a_{n+2}}=\frac{(n+3) a_{n}}{4 a_{n+1} a_{n+2}}
$$
holds true for every positive integer $n \geq 2$.