How should the arithmetic progression be...?

Question

How should the arithmetic progression be, in order that sum of every 2 numbers (which are part of the progression) is part of the progression?
The same question about the geometric progression.

Answer 1

All numbers in the progression are part of the progression.
Are you saying that the sum of ANY two numbers in the progression, or ANY two consecutive numbers in the progression?

2, 4, 6, 8, 10+....+ 2n. This arithmetic progression would seem to satisfy your requirement.

1, r , r^2, r^3, r^4 ... r^n is a geometric progression. Here I guess you want to find r such that r^2+r^3 = r^k and r^5+r^6 = r^j. I don't think this is possible.