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n >= 2^{k-1} for k terms in geometric sequence of an arithmetic sequence

Source: 2006 VMEO III Shortlist SL N2 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

August 30, 2022
geometric sequencenumber theoryarithmetic sequence

Problem Statement

Let a1,a2,...a_1,a_2,... be an arithmetic sequence with the common difference between terms is positive. Assume there are kk terms of this sequence creates an geometric sequence with common ratio dd. Prove that n≥2k−1n\ge 2^{k-1}.
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