Problem #PRU-66111

Problems Calculus Number sequences Boundedness, monotonicity Methods Examples and counterexamples

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?