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Two sequences are equal

Source: Russian TST 2019, Day 7 P1 (Group NG), P2 (Groups A & B)

March 27, 2023
algebraSequences

Problem Statement

Let a0,a1,…,ana_0, a_1, \ldots , a_n and b0,b1,…,bnb_0, b_1, \ldots , b_n be sequences of real numbers such that a0=b0⩾0a_0 = b_0 \geqslant 0, an=bn>0a_n = b_n > 0 and a_i=\sqrt{\frac{a_{i+1}+a_{i-1}}{2}},  b_i=\sqrt{\frac{b_{i+1}+b_{i-1}}{2}},for all i=1,…,n−1i=1,\ldots,n-1. Prove that a1=b1a_1 = b_1.
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