Need help solving a recurrence definition in combinatorics:

Question

anonymous · Answer

Passwords for a certain computer system are strings of uppercase letters. A valid password must contain an even number of X's. Determine a recurrence relation for the number of valid passwords of length n.

anonymous · Answer

adding an x to a bad password would work, adding any other letter to a bad password would work

fwizbang · Answer

let a_n be the no of good pass words with n letters and b_n be the number of bad passwords with n letters.

a_n+b_n = (26)^n

a_(n+1) = 25*a_n + b_n  
b_(n+1)=       a_n  +25 b_n ?

anonymous · Answer

is it legal to define this way, or are we only allowed one variable

fwizbang · Answer

you can always eliminate b_n using the first equation

anonymous · Answer

how would that look if you did it eliminating b_n