MathDB

Consider a sequence of words , consisting of the letters

A

and

B

. The first word in the sequence is "

A

" . The k-th word i s obtained from the

(k-1)

-th by means of the following transformation : each

A

is substituted by

AAB

, and each

B

is substituted by

A

. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters

AABAABAAA BAABAAB...

(a) In which place of this sequence is the

1000

-th letter

A

? (b ) Prove that this sequence is not periodic.
(V . Galperin , Moscows)

Problem Statement