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infinitely many powers of 2, a_{n+1} = a_n + b_n

Source: Indian Postal Coaching 2009 set 3 p1

May 26, 2020
Sequencepower of 2recurrence relationalgebra

Problem Statement

Let a1,a2,a3,...,an,...a_1, a_2, a_3, . . . , a_n, . . . be an infinite sequence of natural numbers in which a1a_1 is not divisible by 55. Suppose an+1=an+bna_{n+1} = a_n + b_n where bn is the last digit of ana_n, for every nn. Prove that the sequence {an}\{a_n\} contains infinitely many powers of 2.
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