Question 89
Define Fibonacci sequence $\{F\}_{n=0}^{\infty}$ as $F_{0}=0$, $F_{1}=1$ and $F_{n+1}=F_{n}+ F_{n-1}$ for every integer $n\gt 1$. Determine all quadruples $(a, b, c, n)$ of positive integers with $a\lt b\lt c$ such that each of $a, b, c, a+n, b+n, c+2n$ is a term of the Fibonacci sequence.