MathDB

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Part of 2012 Benelux

Problems(1)

When the sequence has infinitely many powers of 2

Source: Benelux Mathematical Olympiad 2012

4/23/2012
A sequence a1,a2,…,an,…a_1,a_2,\ldots ,a_n,\ldots of natural numbers is defined by the rule an+1=an+bn (n=1,2,…)a_{n+1}=a_n+b_n\ (n=1,2,\ldots) where bnb_n is the last digit of ana_n. Prove that such a sequence contains infinitely many powers of 22 if and only if a1a_1 is not divisible by 55.
floor functionarithmetic sequencenumber theory proposednumber theory