Question

Consider $a_{n+1} = \frac{1}{1 + \frac{1}{a_n}}$ for $n = 1, 2, \dots, 2008, 2009$ where $a_1 = 1$. Find the value of $a_1 a_2 + a_2 a_3 + a_3 a_4 + \dots + a_{2008} a_{2009}$.

Accepted Answer

C. $a_1 = 1$, $a_2 = 1/2$, $a_3 = 1/3, \dots, a_n = 1/n$. $a_n a_{n+1} = \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$. Sum = $(1 - 1/2) + (1/2 - 1/3) + \dots + (1/2008 - 1/2009) = 1 - \frac{1}{2009} = \frac{2008}{2009}$.

Sequence Product Telescoping Sum

Explanation

Practise this under exam conditions