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Integer trapped between two sequences

High school · ●●●○○ 50 pts

Let {an}\{a_n\} and {bn}\{b_n\} be sequences with a1=0a_1=0 and b1=2013b_1=2013, such that for every positive integer nn the triples (an,an+1,bn)(a_n, a_{n+1}, b_n) and (an+1,bn+1,bn)(a_{n+1}, b_{n+1}, b_n) are arithmetic sequences. Find the unique positive integer cc such that an<c<bna_n < c < b_n for all positive integers nn.

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recurrence relationsgeometric sequences