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non-constant arithmetic progression a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}

Source: St. Petersburg 2019 10.1

May 1, 2019
arithmetic sequencealgebraArithmetic ProgressionRecurrencerecurrence relation

Problem Statement

For a non-constant arithmetic progression (an)(a_n) there exists a natural nn such that an+an+1=a1+…+a3n−1a_{n}+a_{n+1} = a_{1}+…+a_{3n-1} . Prove that there are no zero terms in this progression.
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