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Boundedness, monotonicity , Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Identical transformations , Sequnces

Prove that for any natural number $a_1> 1$ there exists an increasing sequence of natural numbers $a_1, a_2, a_3$, …, for which $a_1^2+ a_2^2 +…+ a_k^2$ is divisible by $a_1+ a_2+…+ a_k$ for all k ≥ 1.

Sequnces

Find the missing numbers:

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