MathDB

From given positive numbers, the following infinite sequence is defined:

a_1

is the sum of all original numbers,

a_2

is the sum of the squares of all original numbers,

a_3

is the sum of the cubes of all original numbers, and so on (

a_k

is the sum of the

k

-th powers of all original numbers). a) Can it happen that

a_1 > a_2 > a_3 > a_4 > a_5

and

a_5 < a_6 < a_7 < \ldots

? (4 points) b) Can it happen that

a_1 < a_2 < a_3 < a_4 < a_5

and

a_5 > a_6 > a_7 > \ldots

? (4 points)
(Alexey Tolpygo)

Problem Statement