Let a_n and b_n be the number of A's and B's respectively in the nth word and f_n the total number of letters in the nth word. Note that in each word there is an A for every B in the previous word so

an = bn-1    (1).

The number of B's is given by the number of B's in the previous word plus the number of A's in the previous word and so

bn+1 = an + bn    (2)

. Putting these two expressions together and substituting for a_n in (2) we get

bn+1 = bn-1+ bn

so the sequence b_n is a Fibonacci sequence and the pattern will continue. Similarly substituting for b_n in (2) we get

an+2=an + an+1

so the a_n forn a Fibonaci sequence and the pattern will continue.

Because the two sequences of numbers are the same apart from the shift of one place the total number of letters is also a Fibonacci sequence and the pattern will continue.

an = bn-1 fn = an + bn = bn-1 + bn = bn+1 fn-1 = an-1 + bn-1 = an-1 + an = an+1 fn+1 = an+1+ bn+1 = fn-1 + fn .