# Got interested in problems about sequences and thought of another one

We have some positive numbers and a sequence where $a_1$ is the sum of the numbers; $a_2$ is the sum of the squares of the numbers, $a_3$ is the sum of the cubes of the numbers, etc. Could be so that before $a_5$ $a_1<a_2<a_3<a_4 < a_5$ and after $a_5$ $a_5 > a_6 > a_7>...$? (vice verca)

• here are you considering only natural numbers only or can these be real as well – happymath Mar 12 '17 at 16:43
• There can be a lot of positive, real numbers. – student28 Mar 12 '17 at 16:47

No, it is not possible. If any of the numbers is greater than $1$, eventually the powers of it will dominate the sum and higher powers will be greater, so to get the $a_i$ decreasing above $a_5$ you must have all the numbers less than $1$. Then the square of any number less than $1$ is less than the number and $a_1 \gt a_2$