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Prove that a unique constant satifies inequality.

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January 30, 2011
inequalities unsolvedinequalities

Problem Statement

Let {an}0∞\{a_n\}^{\infty}_0 and {bn}0∞\{b_n\}^{\infty}_0 be two sequences determined by the recursion formulas an+1=an+bn,a_{n+1} = a_n + b_n, bn+1=3an+bn,n=0,1,2,⋯ , b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots, and the initial values a0=b0=1a_0 = b_0 = 1. Prove that there exists a uniquely determined constant cc such that n∣can−bn∣<2n|ca_n-b_n| < 2 for all nonnegative integers nn.
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