SEAMO 2027 - Paper D - SET 07 - Self-study Question 8

It is known that $Q_{n+1}=\frac1{1+\frac1{Q_n}}$ for $n=1,2,3,4,\ldots,2017,2018$, with $Q_1=1$. Evaluate $Q_1Q_2+Q_2Q_3+\cdots+Q_{2017}Q_{2018}$.