I am trying to resolve the following recursive relationship

$$ K_ {2i-2} = frac {a + K_ {2i}} {1 + aK_ {2i}} quad; quad K_ {2N} = 0 $$

and $ i = 1, cdots, N $

I want to find the solution for $ K_ {2i} $.

I believe that this relationship needs to be resolved from the top down. My first attempt was to bet $ i = N $ which gives

$$ K_ {2N-2} = a $$

then $ i = N-1 $ gives

$$ K_ {2N-4} = frac {2a} {1 + a ^ 2} $$

then $ i = N-2 $ gives

$$ K_ {2N-6} = frac {a + frac {2a} {1 + a ^ 2}} {1+ frac {2a ^ 2} {1 + a ^ 2}} = frac {a ^ 3 + 3a} {1 + 3a ^ 2} $$

then $ i = N-3 $ gives

$$ K_ {2N-8} = frac {4a ^ 3 + 4a} {1 + 6a ^ 2 + a ^ 4} $$

then $ i = N-4 $ gives

$$ K_ {2N-10} = frac {a ^ 5 + 10a ^ 3 + 5a} {1 + 10a ^ 2 + 5a ^ 4} $$

I can't find a pattern in it. How can I solve this?