CAT 2023 Progression — Previous Year Question

Consider the sequence $a_{n+1} = \dfrac{1}{1 + \dfrac{1}{a_n}}$ for $n = 1, 2, \ldots, 2008, 2009$ where $a_1 = 1$. Find the value of $a_1a_2 + a_2a_3 + a_3a_4 + \ldots + a_{2008}a_{2009}$.