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arithmetic sequence as sum of two geometric sequences

Source: Bulgaria 1973 P2

June 20, 2021
arithmetic sequencealgebraSequencegeometric sequence

Problem Statement

Let the numbers a1,a2,a3,a4a_1,a_2,a_3,a_4 form an arithmetic progression with difference d≠0d\ne0. Prove that there are no exists geometric progressions b1,b2,b3,b4b_1,b_2,b_3,b_4 and c1,c2,c3,c4c_1,c_2,c_3,c_4 such that: a1=b1+c1,a2=b2+c2,a3=b3+c3,a4=b4+c4.a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.
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