>>981559Let's try something different. First note that a_n, b_n are positive for each n.

>>981697>P = 1 + sqrt(2), Q = 1 - sqrt(2)>Also notice that PQ = -1By multiplying the two given equation we get

(PQ)^n = (-1)^n = a_n^2 - 2 b_n^2

Since n in the induction proof is even, we have

a_n^2 = 1 + 2 b_n^2 for even n

a_n/b_n = (1 + 2 b_n^2)/b_n^2 = 1/b_n^2 + 2 for even n

So the question asked is equivalent to show that

1/b_(2k)^2 + 2 > 1/b_(2k+2)^2 + 2

b_(2k)^2 < b_(2k+2)^2

b_(2k) < b_(2k+2),

that is, b_(2k) is monotone increasing, and this can be done by induction or directly with the help of the binomial coefficients.