# Sequence of geometric mean subtracted by arithmetic mean

Let $$a_1,a_2,a_3,\dots$$ be a sequence of positive numbers. Define $$G_n=\sqrt[n]{a_1a_2\dots a_n}~\text{and}~A_n=\frac{a_1+\dots+a_n}{n}.$$ We are supposed to use the result $$u^av^b\leq au+bv \tag{*}$$ for positive numbers $$a,b,u,v$$ with $$a+b=1$$ to show that the sequence $$u_n=n(G_n-A_n)$$ is non-increasing.

First, I formed the inequality $$n(G_n-A_n) \geq (n+1)(G_{n+1}-A_{n+1})$$ with the hope of working backwards and find some way that allows us to use $$(*)$$. But I think I got stuck since I couldn't identify weights that add up to one. Would appreciate some pointers either in identifying the weights or the structure for which $$(*)$$ can be used.

• To go from $n$ to $n+1$ you use something like $u = G_n$ or $A_n$ and $v=a_{n+1}$ with $a=\frac{n}{n+1}$ and $b=\frac1{n+1}$. Note that $G_n \le A_n$. (Incidentally, $a$ and $a_n$ seem to be unrelated and $u$ and $u_n$ seem to be unrelated, which suggests other notation could be clearer) Dec 2, 2022 at 1:32
• Identical duplicate in less than 1 hour: Inequality related to AM-GM? Dec 2, 2022 at 2:11
• Thank you! I got it! Have a good day. Dec 2, 2022 at 2:33