for 7, not sure what you mean by your "simpler sequence" obeying this - i don't see how that's the case. you can do this problem with elementary knowledge, however. as you imply, first transform it into an addition problem from a multiplicative one by taking logs. then just write out the sequence: a, b, a+b, a+2b, 2a+3b, 3a+5b,... and note that the ratio between the coefficients tends to phi alternating above and below.