MathDB
TT2008 Junior A-Level - P7

Source:

September 4, 2010
algebra proposedalgebra

Problem Statement

In an in finite sequence a1,a2,a3,⋯a_1, a_2, a_3, \cdots, the number a1a_1 equals 11, and each an,n>1a_n, n > 1, is obtained from an−1a_{n-1} as follows:
- if the greatest odd divisor of nn has residue 11 modulo 44, then an=an−1+1,a_n = a_{n-1} + 1,
- and if this residue equals 33, then an=an−1−1.a_n = a_{n-1} - 1.
Prove that in this sequence
(a) the number 11 occurs infi nitely many times;
(b) each positive integer occurs infi nitely many times.
(The initial terms of this sequence are 1,2,1,2,3,2,1,2,3,4,3,⋯1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots )
Back to ProblemsView on AoPS