Category: Algebra

Question Number 151609 by mathdanisur last updated on 22/Aug/21 Answered by ghimisi last updated on 22/Aug/21 $$\iff {log}_{xy}{(1+\sqrt{xy})}^2⩾{log}_{\frac{x+y}{2}}(\frac{x+y}{2}+1)\iff$$$$\iff \frac{ln(1+\sqrt{xy})}{ln\sqrt{xy}}⩾\frac{ln(1+\frac{x+y}{2})}{ln\frac{x+y}{2}}(\bullet )$$$${f}\left({t}\right)=\frac{{ln}\left(\mathrm{1}+{t}\right)}{{lnt}},{f}:\left(\mathrm{1};\infty\right)\rightarrow{R} \…

Question Number 151596 by mathdanisur last updated on 22/Aug/21 Answered by dumitrel last updated on 22/Aug/21 $${(\sqrt{a}+\sqrt{b}+\sqrt{c})}^2\stackrel{cbs}{⩽}3(a+b+c)\Rightarrow \sqrt{a}+\sqrt{b}+\sqrt{c}⩽\sqrt{3(a+b+c)}$$$${(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})}^2\stackrel{cbs}{⩽}3(ab+bc+ac)\Rightarrow \sqrt{ab}+\sqrt{bc}+\sqrt{ac}⩽\sqrt{3(ab+bc+ac}$$$$\Rightarrow\left(\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}\right)\left(\sqrt{{ab}}+\sqrt{{bc}}+\sqrt{{ac}}\leqslant\sqrt{\mathrm{3}\left({a}+{b}+{b}\right)\centerdot\mathrm{3}\left({ab}+{bc}+{ac}\right)}=\right.…

Question Number 151573 by amin96 last updated on 22/Aug/21 Answered by Olaf_Thorendsen last updated on 22/Aug/21 $${\mathrm{S}}_n=\underset{k=0}{\sum ^{n-1}}\frac{1}{(3k+1)(3k+2)(3k+3)}$$$$\mathrm{S}_{{n}} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(\frac{\mathrm{1}/\mathrm{2}}{\mathrm{3}{k}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{3}{k}+\mathrm{2}}+\frac{\mathrm{1}/\mathrm{2}}{\mathrm{3}{k}+\mathrm{3}}\right)…

Question Number 20488 by xing last updated on 27/Aug/17 Commented by ajfour last updated on 27/Aug/17 $$Questionis:$$$$Givena=-\sqrt{99}+\sqrt{999}+\sqrt{9999}$$$$b=\sqrt{99}-\sqrt{999}+\sqrt{9999}$$$$c=\sqrt{99}+\sqrt{999}-\sqrt{9999}$$$${Find}\:{the}\:{value}\:{of}…

Question Number 151549 by mathdanisur last updated on 21/Aug/21 $$\mathrm{Compare}:$$$$2^{2^{2^{{.}^{{.}^{.}}}}}\mathrm{and}{3}^{3^{3^{{.}^{{.}^{.}}}}}$$$$\mathrm{Here}\mathrm{it}\mathrm{is}\mathrm{raised}1001\mathrm{times}\mathrm{a}\mathrm{square},$$$$\mathrm{1000}\:\mathrm{times}\:\mathrm{a}\:\mathrm{cube}.…