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a_{n+1} = a_n - b_n and b_{n+1} = 2b_n if a_n >=b_n, or else ...

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

November 21, 2022

algebrarecurrence relationSequences

Problem Statement

The sequences

a_1, a_2, a_3,...

and

b_1, b_2, b_3,...

suffices for all positive integers

n

of the following recursion:

a_{n+1} = a_n - b_n

and

b_{n+1} = 2b_n

, if

a_n \ge b_n

a_{n+1} = 2a_n

and

b_{n+1} = b_n - a_n

, if

a_n < b_n

. For which pairs

(a_1, b_1)

of positive real initial terms is there an index

k

with

a_k = 0