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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.

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  • $\begingroup$ 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) $\endgroup$
    – Henry
    Dec 2, 2022 at 1:32
  • $\begingroup$ Identical duplicate in less than 1 hour: Inequality related to AM-GM? $\endgroup$
    – Ѕᴀᴀᴅ
    Dec 2, 2022 at 2:11
  • $\begingroup$ Thank you! I got it! Have a good day. $\endgroup$
    – KHOOS
    Dec 2, 2022 at 2:33

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