Problem #PRU-66111

Problems Calculus Number sequences Boundedness, monotonicity Methods Examples and counterexamples. Constructive proofs

Problem

We took several positive numbers and constructed the following sequence: $a_{1}$ is the sum of the initial numbers, $a_{2}$ is the sum of the squares of the original numbers, $a_{3}$ is the sum of the cubes of the original numbers, and so on.

a) Could it happen that up to $a_{5}$ the sequence decreases ( $a_{1} > a_{2} > a_{3} > a_{4} > a_{5}$ ), and starting with $a_{5}$ – it increases ( $a_{5} < a_{6} < a_{7} < \dots$ )?

b) Could it be the other way around: before $a_{5}$ the sequence increases, and starting with $a_{5}$ – decreases?

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