Weighted Sums and Rationality

★ 8/10Arithmetic Progression number theory Modular Arithmetic sequences Casework

Problem

Call a positive integer $n$ good if for every arithmetic progression $a_1, a_2, \ldots, a_n$ : whenever

a_1 + 2a_2 + 3a_3 + \cdots + na_n

is rational, at least one of $a_1, a_2, \ldots, a_n$ is rational. Determine, with proof, which of the values $6$ , $8$ , $10$ , $12$ are good.

Answer

Solution

Difficulty	8/10
Topics	Arithmetic Progression, number theory, Modular Arithmetic, sequences, Casework

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