How do I find the term of a recursive sequence? I have $\{a_n\}$ the following sequence:
$a_1 =-1$
$a_k a_{k+1} = - a_k - \frac{1}{4}$
How can I find $a_1 a_2 \cdots a_n$?

$a_{k+1} = -\frac{4a_k+1}{4a_{k}}$
$a_2 = -3/4$
$a_3 = -2/3=-4/6$
$a_4 = -5/8$
$a_5 = -3/5 = -6/10$
$a_6 = -7/12$
So, we prove that $a_n = -(n+1)/(2n)$
Base case: $a_1 = -1$
IH: $a_{k} = -(k+1)/(2k)$
Then, $a_{k+1} = -\frac{4a_k+1}{4a_{k}} = -\frac{-2(k+1)/k + 1}{-2(k+1)/k} = -\frac{-2k-2+k}{-2(k+1)} = -((k+1)+1)/(2(k+1))$
Take it from here.
