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 July 3rd, 2013, 12:10 AM #1 Newbie   Joined: May 2013 Posts: 26 Thanks: 0 Prove that If $a\geq b\geq 0$ then prove that $\sqrt[n]{a^n+b^n}+\sqrt[n+1]{a^{n+1}+b^{n+1}}\leq 2a+b(\sqrt[n]{2}+\sqrt[n+1]{2}-2)$
 July 3rd, 2013, 01:52 AM #2 Senior Member   Joined: Jul 2011 Posts: 118 Thanks: 0 Re: Prove that $1)\,a\ge b \sqrt[n]{a^n+a^n}+\sqrt[n+1]{a^{n+1}+a^{n+1}}\ge\sqrt[n]{a^n+b^n}+\sqrt[n+1]{a^{n+1}+b^{n+1}}\ge \sqrt[n]{b^n+b^n}+\sqrt[n+1]{b^{n+1}+b^{n+1}} a(\sqrt[n]{2}+\sqrt[n+1]{2})\ge\sqrt[n]{a^n+b^n}+\sqrt[n+1]{a^{n+1}+b^{n+1}}\ge b(\sqrt[n]{2}+\sqrt[n+1]{2}) 2)\,a(\sqrt[n]{2}+\sqrt[n+1]{2})-b(\sqrt[n]{2}+\sqrt[n+1]{2})=(\sqrt[n]{2}+\sqrt[n+1]{2})(a-b)\ge 2(a-b) 3)\,b(\sqrt[n]{2}+\sqrt[n+1]{2})+2(a-b)=2a+b(\sqrt[n]{2}+\sqrt[n+1]{2}-2)\ge\sqrt[n]{a^n+b^n}+\sqrt[n+1]{a^{n+1}+b^{n+1}}$

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