The sequence (−1),(−1)(−1)1,(−1)(−1)(−1)11,… clearly converges to the integer −1.
The sequence (4),(4)(4)1,(4)(4)(4)11,… converges to the integer 2. (Can you prove it?)
Does there exist another value b=1,−1,4 such that the sequence {an} defined recursively by
a0=b;an+1=ban1
also converges to an integer?
Note: Written out, this sequence is (b),(b)(b)1,(b)(b)(b)11,….