Mathematical Girls - Vol. 2 Ch. 9

[MangaDex]

 

[Bumpf]

can this be expanded n real numbers (k_1 to k_n) rather than two? That is, the arithmetic mean (sum of all of them over n) always being more than their geometric mean (product of all of them to the 1/n power) ?

 

[syketuri]

can this be expanded n real numbers (k_1 to k_n) rather than two? That is, the arithmetic mean (sum of all of them over n) always being more than their geometric mean (product of all of them to the 1/n power) ?

You probably got an answer to this within the year and a half of me answering, but yes! In fact, what you are considering as the “generalized” AM-GM is actually its regular form! They even have weigted versions where we take linear combinations of these k_n’s with some given reals w_n, provided the sum w = w_1 + … + w_n > 0. Here’s the wiki:
https://en.m.wikipedia.org/wiki/AM–GM_inequality

 

[Bumpf]

You probably got an answer to this within the year and a half of me answering, but yes! In fact, what you are considering as the “generalized” AM-GM is actually its regular form! They even have weigted versions where we take linear combinations of these k_n’s with some given reals w_n, provided the sum w = w_1 + … + w_n > 0. Here’s the wiki:
https://en.m.wikipedia.org/wiki/AM–GM_inequality

Oh wow that's really cool ) I haven't really read much into the AM-GM inequality, I didn't know about that! Thanks ))))