Relationship between P(0,n) and P(0,n+1) in BDTs

I want to understand the generalized relationship between P(0,n) and P(0,n+1) in BDTs.

One problem, we could solve:
P(0,2) = P(0,1)*.5( (1/(1+r_u)^n) + (1/(1+r_d)^n))

But I tried to generalize this to solve for P(0,3):
P(0,3) = P(0,2) *.5( (1/(1+r_u)^n) + (1/(1+r_d)^n))
and this was wrong.

Can someone please help!


Relationship between P(0,n) and P(0,n+1) in BDTs