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c^2_r=c_kc_m, c_n=a_n+b_n,a_n =9a_{n-1}-2b_{n-1}, b_n=2a_{n-1}+4b_{n-1},

Source: 2012 Balkan Shortlist BMO N2

April 5, 2020
recurrence relationnumber theorySequencenumber theory with sequences

Problem Statement

Let the sequences (an)n=1∞(a_n)_{n=1}^{\infty} and (bn)n=1∞(b_n)_{n=1}^{\infty} satisfy a0=b0=1,an=9an−1−2bn−1a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1} and bn=2an−1+4bn−1b_n = 2a_{n-1} + 4b_{n-1} for all positive integers nn. Let cn=an+bnc_n = a_n + b_n for all positive integers nn. Prove that there do not exist positive integers k,r,mk, r, m such that cr2=ckcmc^2_r = c_kc_m.
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